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InfraredAbsorptionByTheAtmosphere.html
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Plot global temperature anomalies from Berkeley Earth
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Calculate Percentage of Radiation in Spectral Bands
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Insolation
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Infrared Absorption by the Atmosphere
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Calculation of the Equilibrium Temperature of Earth
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Exponential Growth
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Definition of the Optical Depth
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Optical Depth in the Atmosphere
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The McClatchey Standard Atmospheres
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1) Choose the standard atmosphere
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2) Set the atmospheric CO2 concentration (in ppmv)
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Calculating Optical Depth
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Calculating Atmospheric Transmission
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Calculating Atmospheric Absorption and Emissivity
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The McClatchey Standard Atmospheres
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2) Set the atmospheric CO2 concentration (in ppmv)
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Primary Greenhouse Gas Concentrations in Standard Atmospheres
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Calculating Optical Depth
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Calculating Atmospheric Transmission
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<section class="tex2jax_ignore mathjax_ignore" id="infrared-absorption-by-the-atmosphere">
<h1>Infrared Absorption by the Atmosphere<a class="headerlink" href="#infrared-absorption-by-the-atmosphere" title="Permalink to this headline">#</a></h1>
<p><i>© Von P. Walden, Washington State University</i></p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
</pre></div>
</div>
</div>
</div>
<section id="definition-of-the-optical-depth">
<h2>Definition of the Optical Depth<a class="headerlink" href="#definition-of-the-optical-depth" title="Permalink to this headline">#</a></h2>
<p>To calculate the infrared absorption by the atmosphere, we must calculate the optical depth. The optical depth is related to the transmission of radiation through “optical media”, such as an absorbing gas or an absorbing liquid. <a class="reference external" href="https://en.wikipedia.org/wiki/Beer%E2%80%93Lambert_law">Beer’s Law</a> gives us</p>
<div class="math notranslate nohighlight">
\[ I(\lambda) = I_o(\lambda) e^{-\tau} \]</div>
<p>This can be rewritten as</p>
<div class="math notranslate nohighlight">
\[ Transmission = \frac{I(\lambda)}{I_o(\lambda)} = e^{-\tau} \]</div>
<p>Then one can solve for the optical depth <span class="math notranslate nohighlight">\(\tau\)</span> as</p>
<div class="math notranslate nohighlight">
\[ \tau = -ln \Big[ \frac{I(\lambda)}{I_o(\lambda)} \Big] \]</div>
<p>In general, the optical depth is:</p>
<div class="math notranslate nohighlight">
\[ \tau = \int_{z_1}^{z_2} \beta_{\lambda} dz = \int_{z_1}^{z_2} \kappa_{\lambda} \rho (z) dz = \int_{z_1}^{z_2} \sigma_{\lambda} n(z) dz \]</div>
<p>where</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(\beta\)</span> is the volume absorption coefficient,</p></li>
<li><p><span class="math notranslate nohighlight">\(\kappa\)</span> is the mass absorption coefficient,</p></li>
<li><p><span class="math notranslate nohighlight">\(\sigma\)</span> is the absorption cross section,</p></li>
</ul>
<p>which all depend on wavelength, <span class="math notranslate nohighlight">\(\lambda\)</span>.</p>
<p><span class="math notranslate nohighlight">\(\rho(z)\)</span> and n(z) are the air density and the number density of the absorbing gas. z is distance along the absorbing path from <span class="math notranslate nohighlight">\(z_1\)</span> to <span class="math notranslate nohighlight">\(z_2\)</span>.</p>
<p>If one conducts an experiment of Beer’s Law in the lab using a tube filled with an absorbing gas (like H2O or CO2), this equation reduces to:</p>
<div class="math notranslate nohighlight">
\[ \tau = \beta (z_2 - z_1) = \kappa_{\lambda} \rho (z_2 - z_1) = \sigma_{\lambda} N (z_2 - z_1) \]</div>
<p>Note that this is how scientists (100 or so years ago) determined how different gases absorbed infrared radiation at different wavelengths, by measuring the transmission of radiation through the gas, calculating the optical depth, then solving for either <span class="math notranslate nohighlight">\(\beta_{\lambda}\)</span>, <span class="math notranslate nohighlight">\(\kappa_{\lambda}\)</span>, or <span class="math notranslate nohighlight">\(\sigma_{\lambda}\)</span>.</p>
<section id="optical-depth-in-the-atmosphere">
<h3>Optical Depth in the Atmosphere<a class="headerlink" href="#optical-depth-in-the-atmosphere" title="Permalink to this headline">#</a></h3>
<p>To determine the optical depth in the atmosphere, one must use the general equation above, because the number of molecules of absorbing gas changes with height in the atmosphere. Carbon dioxide is “uniformly mixed” in the atmosphere, meaning that its concentration relative to other gases does not change with height. However, the actual number of CO2 molecules decreases exponentially with height in the atmosphere because the density (and pressure) of the air decreases in this manner. Most of the water vapor in the atmosphere is contained only in the troposphere in the lowest 7 or 8 km. Therefore, one must integrate through the atmosphere to determine the optical depth using:</p>
<div class="math notranslate nohighlight">
\[ \tau = \int_{z_1}^{z_2} \kappa_{\lambda} \rho (z) dz \]</div>
<p>Here we have chosen to use the specific formula of the optical depth that contains the mass absorption coefficient, because Figure 2 of Pierrehumbert (2010) gives values of <span class="math notranslate nohighlight">\(\kappa\)</span> for both H2O and CO2.</p>
</section>
</section>
<section id="the-mcclatchey-standard-atmospheres">
<h2>The McClatchey Standard Atmospheres<a class="headerlink" href="#the-mcclatchey-standard-atmospheres" title="Permalink to this headline">#</a></h2>
<p>We’re lucky because <a class="reference external" href="https://apps.dtic.mil/sti/pdfs/AD0753075.pdf">McClatchey et al (1972)</a> [in Optical Properties of the Atmosphere (Third Edition)] created a series of “standard atmospheres” for different locations and seasons on Earth that give us n(z) as a function of altitude (z) in the atmosphere. The McClatchey Standard Atmospheres are for:</p>
<ul class="simple">
<li><p>Tropics (TRP)</p></li>
<li><p>Mid-latitude Summer (MLS)</p></li>
<li><p>Mid-latitude Winter (MLW)</p></li>
<li><p>Sub-Arctic Summer (SAS)</p></li>
<li><p>Sub-Arctic Winter (SAW</p></li>
</ul>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Read in raw data files</span>
<span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'layer'</span><span class="p">,</span> <span class="s1">'altitude'</span><span class="p">,</span> <span class="s1">'pressure'</span><span class="p">,</span> <span class="s1">'temperature'</span><span class="p">,</span> <span class="s1">'dew_point'</span><span class="p">,</span> <span class="s1">'density'</span><span class="p">,</span> <span class="s1">'h2o'</span><span class="p">,</span> <span class="s1">'co2'</span><span class="p">,</span> <span class="s1">'o3'</span><span class="p">,</span> <span class="s1">'n2o'</span><span class="p">,</span> <span class="s1">'co'</span><span class="p">,</span> <span class="s1">'ch4'</span><span class="p">,</span> <span class="s1">'o2'</span><span class="p">]</span>
<span class="n">trp</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_table</span><span class="p">(</span><span class="s1">'../trp.dat'</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">'\s+'</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="n">columns</span><span class="p">)</span>
<span class="n">mls</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_table</span><span class="p">(</span><span class="s1">'../mls.dat'</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">'\s+'</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="n">columns</span><span class="p">)</span>
<span class="n">mlw</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_table</span><span class="p">(</span><span class="s1">'../mlw.dat'</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">'\s+'</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="n">columns</span><span class="p">)</span>
<span class="n">sas</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_table</span><span class="p">(</span><span class="s1">'../sas.dat'</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">'\s+'</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="n">columns</span><span class="p">)</span>
<span class="n">saw</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_table</span><span class="p">(</span><span class="s1">'../saw.dat'</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s1">'\s+'</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="n">columns</span><span class="p">)</span>
</pre></div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Rair</span> <span class="o">=</span> <span class="mf">287.058</span> <span class="c1"># J kg-1 K-1</span>
<span class="c1"># Calculate air density</span>
<span class="n">trp</span><span class="o">.</span><span class="n">rename</span><span class="p">({</span><span class="s1">'density'</span><span class="p">:</span> <span class="s1">'numberDensity'</span><span class="p">})</span>
<span class="n">trp</span><span class="p">[</span><span class="s1">'airDensity'</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">trp</span><span class="o">.</span><span class="n">pressure</span> <span class="o">*</span> <span class="mi">100</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">Rair</span> <span class="o">*</span> <span class="n">trp</span><span class="o">.</span><span class="n">temperature</span><span class="p">)</span>
<span class="n">mls</span><span class="o">.</span><span class="n">rename</span><span class="p">({</span><span class="s1">'density'</span><span class="p">:</span> <span class="s1">'numberDensity'</span><span class="p">})</span>
<span class="n">mls</span><span class="p">[</span><span class="s1">'airDensity'</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">mls</span><span class="o">.</span><span class="n">pressure</span> <span class="o">*</span> <span class="mi">100</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">Rair</span> <span class="o">*</span> <span class="n">mls</span><span class="o">.</span><span class="n">temperature</span><span class="p">)</span>
<span class="n">mlw</span><span class="o">.</span><span class="n">rename</span><span class="p">({</span><span class="s1">'density'</span><span class="p">:</span> <span class="s1">'numberDensity'</span><span class="p">})</span>
<span class="n">mlw</span><span class="p">[</span><span class="s1">'airDensity'</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">mlw</span><span class="o">.</span><span class="n">pressure</span> <span class="o">*</span> <span class="mi">100</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">Rair</span> <span class="o">*</span> <span class="n">mlw</span><span class="o">.</span><span class="n">temperature</span><span class="p">)</span>
<span class="n">sas</span><span class="o">.</span><span class="n">rename</span><span class="p">({</span><span class="s1">'density'</span><span class="p">:</span> <span class="s1">'numberDensity'</span><span class="p">})</span>
<span class="n">sas</span><span class="p">[</span><span class="s1">'airDensity'</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">sas</span><span class="o">.</span><span class="n">pressure</span> <span class="o">*</span> <span class="mi">100</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">Rair</span> <span class="o">*</span> <span class="n">sas</span><span class="o">.</span><span class="n">temperature</span><span class="p">)</span>
<span class="n">saw</span><span class="o">.</span><span class="n">rename</span><span class="p">({</span><span class="s1">'density'</span><span class="p">:</span> <span class="s1">'numberDensity'</span><span class="p">})</span>
<span class="n">saw</span><span class="p">[</span><span class="s1">'airDensity'</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">saw</span><span class="o">.</span><span class="n">pressure</span> <span class="o">*</span> <span class="mi">100</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">Rair</span> <span class="o">*</span> <span class="n">saw</span><span class="o">.</span><span class="n">temperature</span><span class="p">)</span>
</pre></div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Interpolate the standard atmospheres to a common grid in meters with heights every 100 meters</span>
<span class="n">trp</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">trp</span><span class="o">.</span><span class="n">altitude</span><span class="o">*</span><span class="mi">1000</span>
<span class="n">trp</span> <span class="o">=</span> <span class="n">trp</span><span class="o">.</span><span class="n">reindex</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">60001</span><span class="p">,</span><span class="mi">100</span><span class="p">))</span><span class="o">.</span><span class="n">interpolate</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">'linear'</span><span class="p">)</span>
<span class="n">mls</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">mls</span><span class="o">.</span><span class="n">altitude</span><span class="o">*</span><span class="mi">1000</span>
<span class="n">mls</span> <span class="o">=</span> <span class="n">mls</span><span class="o">.</span><span class="n">reindex</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">60001</span><span class="p">,</span><span class="mi">100</span><span class="p">))</span><span class="o">.</span><span class="n">interpolate</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">'linear'</span><span class="p">)</span>
<span class="n">mlw</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">mlw</span><span class="o">.</span><span class="n">altitude</span><span class="o">*</span><span class="mi">1000</span>
<span class="n">mlw</span> <span class="o">=</span> <span class="n">mlw</span><span class="o">.</span><span class="n">reindex</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">60001</span><span class="p">,</span><span class="mi">100</span><span class="p">))</span><span class="o">.</span><span class="n">interpolate</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">'linear'</span><span class="p">)</span>
<span class="n">sas</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">sas</span><span class="o">.</span><span class="n">altitude</span><span class="o">*</span><span class="mi">1000</span>
<span class="n">sas</span> <span class="o">=</span> <span class="n">sas</span><span class="o">.</span><span class="n">reindex</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">60001</span><span class="p">,</span><span class="mi">100</span><span class="p">))</span><span class="o">.</span><span class="n">interpolate</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">'linear'</span><span class="p">)</span>
<span class="n">saw</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">saw</span><span class="o">.</span><span class="n">altitude</span><span class="o">*</span><span class="mi">1000</span>
<span class="n">saw</span> <span class="o">=</span> <span class="n">saw</span><span class="o">.</span><span class="n">reindex</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">60001</span><span class="p">,</span><span class="mi">100</span><span class="p">))</span><span class="o">.</span><span class="n">interpolate</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">'linear'</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
</section>
<hr class="docutils" />
<section id="choose-the-standard-atmosphere">
<h2>1) Choose the standard atmosphere<a class="headerlink" href="#choose-the-standard-atmosphere" title="Permalink to this headline">#</a></h2>
<table class="colwidths-auto table">
<thead>
<tr class="row-odd"><th class="head"><p>Standard Atmosphere</p></th>
<th class="head"><p>stdatm_name</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Tropical</p></td>
<td><p>‘trp’</p></td>
</tr>
<tr class="row-odd"><td><p>Mid-Latitude Summer</p></td>
<td><p>‘mls’</p></td>
</tr>
<tr class="row-even"><td><p>Mid-Latitude Winter</p></td>
<td><p>‘mlw’</p></td>
</tr>
<tr class="row-odd"><td><p>Sub-Arctic Summer</p></td>
<td><p>‘sas’</p></td>
</tr>
<tr class="row-even"><td><p>Sub-Arctic Winter</p></td>
<td><p>‘saw’</p></td>
</tr>
</tbody>
</table>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Change this variable to choose the model atmosphere</span>
<span class="c1"># !! NOTE that the single quotation marks are important !!</span>
<span class="n">stdatm_name</span> <span class="o">=</span> <span class="s1">'mlw'</span>
<span class="k">if</span> <span class="s1">'trp'</span> <span class="ow">in</span> <span class="n">stdatm_name</span><span class="p">:</span>
<span class="n">stdatm</span> <span class="o">=</span> <span class="n">trp</span>
<span class="k">elif</span> <span class="s1">'mls'</span> <span class="ow">in</span> <span class="n">stdatm_name</span><span class="p">:</span>
<span class="n">stdatm</span> <span class="o">=</span> <span class="n">mls</span>
<span class="k">elif</span> <span class="s1">'mlw'</span> <span class="ow">in</span> <span class="n">stdatm_name</span><span class="p">:</span>
<span class="n">stdatm</span> <span class="o">=</span> <span class="n">mlw</span>
<span class="k">elif</span> <span class="s1">'sas'</span> <span class="ow">in</span> <span class="n">stdatm_name</span><span class="p">:</span>
<span class="n">stdatm</span> <span class="o">=</span> <span class="n">sas</span>
<span class="k">elif</span> <span class="s1">'saw'</span> <span class="ow">in</span> <span class="n">stdatm_name</span><span class="p">:</span>
<span class="n">stdatm</span> <span class="o">=</span> <span class="n">saw</span>
<span class="k">else</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"ERROR: This standard atmosphere is NOT RECOGNIZED. Try setting stdatm_name again!!"</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
</section>
<section id="set-the-atmospheric-co2-concentration-in-ppmv">
<h2>2) Set the atmospheric CO2 concentration (in ppmv)<a class="headerlink" href="#set-the-atmospheric-co2-concentration-in-ppmv" title="Permalink to this headline">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set CO2 in all Standard Atmospheres.</span>
<span class="c1">#co2 = 330. # ppmv; approximate concentration in 1971</span>
<span class="c1">#co2 = 419. # ppmv; current concentration in 2023</span>
<span class="c1">#co2 = 280. # ppmv; pre-industrial concentration</span>
<span class="n">co2</span> <span class="o">=</span> <span class="mf">560.</span> <span class="c1"># ppmv; doubled CO2 concentration (from pre-industrial)</span>
<span class="n">trp</span><span class="p">[</span><span class="s1">'co2'</span><span class="p">]</span> <span class="o">=</span> <span class="n">mls</span><span class="p">[</span><span class="s1">'co2'</span><span class="p">]</span> <span class="o">=</span> <span class="n">mlw</span><span class="p">[</span><span class="s1">'co2'</span><span class="p">]</span> <span class="o">=</span> <span class="n">sas</span><span class="p">[</span><span class="s1">'co2'</span><span class="p">]</span> <span class="o">=</span> <span class="n">saw</span><span class="p">[</span><span class="s1">'co2'</span><span class="p">]</span> <span class="o">=</span> <span class="n">co2</span>
</pre></div>
</div>
</div>
</div>
<hr class="docutils" />
<section id="pressure-temperature-and-air-density-in-the-standard-atmospheres">
<h3>Pressure, Temperature, and Air Density in the Standard Atmospheres<a class="headerlink" href="#pressure-temperature-and-air-density-in-the-standard-atmospheres" title="Permalink to this headline">#</a></h3>
<div class="cell tag_hide-input docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">,</span> <span class="n">ax3</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># Pressure</span>
<span class="n">trp</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'pressure'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mls</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'pressure'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mlw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'pressure'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">sas</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'pressure'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">saw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'pressure'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Pressure (mb)'</span><span class="p">);</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Altitude (km)'</span><span class="p">);</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Atmospheric Pressure for Standard Atmospheres'</span><span class="p">);</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'TRP'</span><span class="p">,</span><span class="s1">'MLS'</span><span class="p">,</span><span class="s1">'MLW'</span><span class="p">,</span><span class="s1">'SAS'</span><span class="p">,</span><span class="s1">'SAW'</span><span class="p">]);</span>
<span class="c1"># Temperature</span>
<span class="n">trp</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'temperature'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mls</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'temperature'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mlw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'temperature'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">sas</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'temperature'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">saw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'temperature'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Temperature (K)'</span><span class="p">);</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Altitude (km)'</span><span class="p">);</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Temperature Profiles for Standard Atmospheres'</span><span class="p">);</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'TRP'</span><span class="p">,</span><span class="s1">'MLS'</span><span class="p">,</span><span class="s1">'MLW'</span><span class="p">,</span><span class="s1">'SAS'</span><span class="p">,</span><span class="s1">'SAW'</span><span class="p">]);</span>
<span class="c1">#O3</span>
<span class="n">trp</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'airDensity'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mls</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'airDensity'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mlw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'airDensity'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">sas</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'airDensity'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">saw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'airDensity'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Density (kg m-3)'</span><span class="p">);</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Altitude (km)'</span><span class="p">);</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Density of Air for Standard Atmospheres'</span><span class="p">);</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'TRP'</span><span class="p">,</span><span class="s1">'MLS'</span><span class="p">,</span><span class="s1">'MLW'</span><span class="p">,</span><span class="s1">'SAS'</span><span class="p">,</span><span class="s1">'SAW'</span><span class="p">]);</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="../_images/InfraredAbsorptionByTheAtmosphere_19_0.png" src="../_images/InfraredAbsorptionByTheAtmosphere_19_0.png" />
</div>
</div>
</section>
<section id="primary-greenhouse-gas-concentrations-in-standard-atmospheres">
<h3>Primary Greenhouse Gas Concentrations in Standard Atmospheres<a class="headerlink" href="#primary-greenhouse-gas-concentrations-in-standard-atmospheres" title="Permalink to this headline">#</a></h3>
<div class="cell tag_hide-input docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="p">(</span><span class="n">ax1</span><span class="p">,</span> <span class="n">ax2</span><span class="p">,</span> <span class="n">ax3</span><span class="p">)</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span><span class="mi">10</span><span class="p">),</span> <span class="n">nrows</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">ncols</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># H2O</span>
<span class="n">trp</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'h2o'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mls</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'h2o'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mlw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'h2o'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">sas</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'h2o'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">saw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax1</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'h2o'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Water Vapor (ppmv)'</span><span class="p">);</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Altitude (km)'</span><span class="p">);</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Water Vapor concentration for Standard Atmospheres'</span><span class="p">);</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'TRP'</span><span class="p">,</span><span class="s1">'MLS'</span><span class="p">,</span><span class="s1">'MLW'</span><span class="p">,</span><span class="s1">'SAS'</span><span class="p">,</span><span class="s1">'SAW'</span><span class="p">]);</span>
<span class="c1"># CO2</span>
<span class="n">trp</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'co2'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mls</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'co2'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mlw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'co2'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">sas</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'co2'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">saw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax2</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'co2'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Carbon Dioxide (ppmv)'</span><span class="p">);</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Altitude (km)'</span><span class="p">);</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Carbon Dioxide concentration for Standard Atmospheres'</span><span class="p">);</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'TRP'</span><span class="p">,</span><span class="s1">'MLS'</span><span class="p">,</span><span class="s1">'MLW'</span><span class="p">,</span><span class="s1">'SAS'</span><span class="p">,</span><span class="s1">'SAW'</span><span class="p">]);</span>
<span class="c1">#O3</span>
<span class="n">trp</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'o3'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mls</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'o3'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">mlw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'o3'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">sas</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'o3'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">saw</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax3</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'o3'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">)</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Ozone (ppmv)'</span><span class="p">);</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Altitude (km)'</span><span class="p">);</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Ozone concentration for Standard Atmospheres'</span><span class="p">);</span>
<span class="n">ax3</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'TRP'</span><span class="p">,</span><span class="s1">'MLS'</span><span class="p">,</span><span class="s1">'MLW'</span><span class="p">,</span><span class="s1">'SAS'</span><span class="p">,</span><span class="s1">'SAW'</span><span class="p">]);</span>
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</section>
<section id="calculating-optical-depth">
<h2>Calculating Optical Depth<a class="headerlink" href="#calculating-optical-depth" title="Permalink to this headline">#</a></h2>
<p>Now that we have values for the air density and the concentrations of the greenhouse gases, it is easy to calculate the optical depth for individual gases.</p>
<p>We will use mass absorption coefficients for carbon dioxide at the following wavelengths (from the center of the 15-um CO2 band towards the atmospheric window between 8 and 12.5 um):</p>
<table class="colwidths-auto table">
<thead>
<tr class="row-odd"><th class="head"><p>Type of Absorption</p></th>
<th class="head"><p>Wavenumber (<span class="math notranslate nohighlight">\(cm^{-1}\)</span>)</p></th>
<th class="head"><p>Wavelength (<span class="math notranslate nohighlight">\(\mu\)</span>m)</p></th>
<th class="head"><p><span class="math notranslate nohighlight">\(\kappa_a\)</span> (<span class="math notranslate nohighlight">\(m^2 kg^{-1}\)</span>)</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p>Very Strong</p></td>
<td><p>667</p></td>
<td><p>15</p></td>
<td><p>2000</p></td>
</tr>
<tr class="row-odd"><td><p>Very Strong</p></td>
<td><p>714</p></td>
<td><p>14</p></td>
<td><p>10</p></td>
</tr>
<tr class="row-even"><td><p>Moderate</p></td>
<td><p>769</p></td>
<td><p>13</p></td>
<td><p>0.1</p></td>
</tr>
<tr class="row-odd"><td><p>Weak</p></td>
<td><p>833</p></td>
<td><p>12</p></td>
<td><p>0.001</p></td>
</tr>
</tbody>
</table>
<p>These mass absorption coefficients values were <em>approximated</em> from Figure 2 in Pierrehumbert, Physics Today, January 2011.</p>
<p>So, the optical depth can now be calculated for each layer of the atmosphere by multiplying the absorption cross sections (<span class="math notranslate nohighlight">\(\kappa_a\)</span>) by the air density, and then by the thickness of the atmospheric layer, as shown in the code below (for the Mid-latitude Winter Standard Atmosphere - MLW).</p>
<p><strong>Disclaimer:</strong> Please note that this calculation of optical depth is for illustrative purposes only, and is over-simplified in the following ways:</p>
<ul class="simple">
<li><p>In most cases H2O overlaps the absorption bands of all the other GH gases. (See Figure 2 of Pierrehumbert (2010); blowup of 600-670 cm-1).</p></li>
<li><p>We are neglecting how the line strengths of absorption lines depend on temperature in the atmosphere (<a class="reference external" href="https://hitran.org/docs/definitions-and-units/">https://hitran.org/docs/definitions-and-units/</a>)</p></li>
<li><p>We are neglecting how the widths of absorption lines depend on both temperature and pressure (<a class="reference external" href="https://hitran.org/docs/definitions-and-units/">https://hitran.org/docs/definitions-and-units/</a>)</p></li>
</ul>
<p>Accurate radiative transfer models of the atmosphere account for all of this issues.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Ka_15</span> <span class="o">=</span> <span class="mi">2000</span>
<span class="n">Ka_14</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">Ka_13</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="n">Ka_12</span> <span class="o">=</span> <span class="mf">0.001</span>
<span class="n">rhoCO2</span> <span class="o">=</span> <span class="n">stdatm</span><span class="o">.</span><span class="n">airDensity</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">stdatm</span><span class="o">.</span><span class="n">co2</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="mf">1e6</span> <span class="c1"># Convert from ppmv to percent fraction</span>
<span class="n">dz</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diff</span><span class="p">(</span><span class="n">stdatm</span><span class="o">.</span><span class="n">altitude</span><span class="p">)</span> <span class="o">*</span> <span class="mi">1000</span> <span class="c1"># Calculate the height of each layer by differencing levels; meters</span>
<span class="c1"># Calculate and plot the optical depth at 14.6 um for CO2 for the MLW Standard Atmosphere.</span>
<span class="n">stdatm</span><span class="p">[</span><span class="s1">'od_15'</span><span class="p">]</span> <span class="o">=</span> <span class="n">Ka_15</span> <span class="o">*</span> <span class="n">rhoCO2</span> <span class="o">*</span> <span class="n">dz</span>
<span class="n">stdatm</span><span class="p">[</span><span class="s1">'od_14'</span><span class="p">]</span> <span class="o">=</span> <span class="n">Ka_14</span> <span class="o">*</span> <span class="n">rhoCO2</span> <span class="o">*</span> <span class="n">dz</span>
<span class="n">stdatm</span><span class="p">[</span><span class="s1">'od_13'</span><span class="p">]</span> <span class="o">=</span> <span class="n">Ka_13</span> <span class="o">*</span> <span class="n">rhoCO2</span> <span class="o">*</span> <span class="n">dz</span>
<span class="n">stdatm</span><span class="p">[</span><span class="s1">'od_12'</span><span class="p">]</span> <span class="o">=</span> <span class="n">Ka_12</span> <span class="o">*</span> <span class="n">rhoCO2</span> <span class="o">*</span> <span class="n">dz</span>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
<span class="n">stdatm</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'od_15'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">logy</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">stdatm</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'od_14'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">logy</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">stdatm</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'od_13'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">logy</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">stdatm</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="s1">'od_12'</span><span class="p">,</span> <span class="n">y</span><span class="o">=</span><span class="s1">'altitude'</span><span class="p">,</span> <span class="n">logx</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">logy</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Optical Depth'</span><span class="p">);</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Altitude (km)'</span><span class="p">);</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s1">'Infrared Optical Depths for CO2 in '</span> <span class="o">+</span> <span class="n">stdatm_name</span> <span class="o">+</span> <span class="s1">'Standard Atmospheres'</span><span class="p">);</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s1">'15 um'</span><span class="p">,</span><span class="s1">'14 um'</span><span class="p">,</span><span class="s1">'13 um'</span><span class="p">,</span><span class="s1">'12 um'</span><span class="p">]);</span>
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</section>
<section id="calculating-atmospheric-transmission">
<h2>Calculating Atmospheric Transmission<a class="headerlink" href="#calculating-atmospheric-transmission" title="Permalink to this headline">#</a></h2>
<section id="as-a-function-of-altitude">
<h3>As a function of altitude<a class="headerlink" href="#as-a-function-of-altitude" title="Permalink to this headline">#</a></h3>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">stdatm</span><span class="p">[</span><span class="s1">'T_15'</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">stdatm</span><span class="o">.</span><span class="n">od_15</span><span class="p">)</span> <span class="o">*</span> <span class="mi">100</span>
<span class="n">stdatm</span><span class="p">[</span><span class="s1">'T_14'</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">stdatm</span><span class="o">.</span><span class="n">od_14</span><span class="p">)</span> <span class="o">*</span> <span class="mi">100</span>