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<div class="section" id="common-formula">
<h1>Common formula</h1>
<div class="section" id="gravity">
<h2>gravity</h2>
<ol class="arabic">
<li><p class="first">Newton’s gravitational law</p>
<table border="1" class="docutils">
<colgroup>
<col width="24%" />
<col width="42%" />
<col width="33%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">description</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(g\)</span></td>
<td>gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g_{surf}\)</span></td>
<td>gravity at earth surface</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(h\)</span></td>
<td>height above surface</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(R_{surf}\)</span></td>
<td>earth radius at surface</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\[g = g_{surf}\left(\frac{R_{surf}}{R_{surf} + h}\right)^2\]</div>
</li>
<li><p class="first">normal gravity at ellipsoid surface</p>
<p>This is the WGS84 ellipsoidal gravity formula as taken from NIMA TR8350.2</p>
<table border="1" class="docutils">
<colgroup>
<col width="28%" />
<col width="36%" />
<col width="36%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(a\)</span></td>
<td>WGS84 semi-major axis</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(b\)</span></td>
<td>WGS84 semi-minor axis</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(e\)</span></td>
<td>eccentricity</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g_{e}\)</span></td>
<td>gravity at equator</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(g_{p}\)</span></td>
<td>gravity at poles</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g_{surf}\)</span></td>
<td>gravity at surface</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\phi\)</span></td>
<td>latitude</td>
<td><span class="math notranslate nohighlight">\(degN\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\begin{eqnarray}
e^2 & = & \frac{a^2-b^2}{a^2} \\
k & = & \frac{bg_{p} - ag_{e}}{ag_{e}} \\
g_{surf} & = & g_{e}\frac{1 + k {\sin}^2(\frac{\pi}{180}\phi)}{\sqrt{1 - e^2{\sin}^2(\frac{\pi}{180}\phi)}} \\
g_{surf} & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)}
{\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}}
\end{eqnarray}</div></li>
<li><p class="first">normal gravity above ellipsoid surface</p>
<p>This is the WGS84 ellipsoidal gravity formula as taken from NIMA TR8350.2</p>
<table border="1" class="docutils">
<colgroup>
<col width="21%" />
<col width="48%" />
<col width="31%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(a\)</span></td>
<td>WGS84 semi-major axis</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(b\)</span></td>
<td>WGS84 semi-minor axis</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(f\)</span></td>
<td>WGS84 flattening</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g\)</span></td>
<td>gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(g_{surf}\)</span></td>
<td>gravity at the ellipsoid surface</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(GM\)</span></td>
<td>WGS84 earth’s gravitational constant</td>
<td><span class="math notranslate nohighlight">\(\frac{m^3}{s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(z\)</span></td>
<td>altitude</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\phi\)</span></td>
<td>latitude</td>
<td><span class="math notranslate nohighlight">\(degN\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\omega\)</span></td>
<td>WGS84 earth angular velocity</td>
<td><span class="math notranslate nohighlight">\(rad/s\)</span></td>
</tr>
</tbody>
</table>
<p>The formula used is the one based on the truncated Taylor series expansion:</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
m & = & \frac{\omega^2a^2b}{GM} \\
g & = & g_{surf} \left[ 1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)z + \frac{3}{a^2}z^2 \right] \\
\end{eqnarray}</div></li>
</ol>
</div>
<div class="section" id="geopotential-height">
<h2>geopotential height</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="24%" />
<col width="42%" />
<col width="33%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">description</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(g\)</span></td>
<td>gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g_{0}\)</span></td>
<td>mean earth gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(g_{surf}\)</span></td>
<td>gravity at earth surface</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(p\)</span></td>
<td>pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(R_{surf}\)</span></td>
<td>earth radius at surface</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(h\)</span></td>
<td>height above surface</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(h_{g}\)</span></td>
<td>geopotential height</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\phi\)</span></td>
<td>latitude</td>
<td><span class="math notranslate nohighlight">\(degN\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\rho\)</span></td>
<td>mass density</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{m^3}\)</span></td>
</tr>
</tbody>
</table>
<p>The geopotential height allows the gravity in the hydrostatic equation</p>
<div class="math notranslate nohighlight">
\[dp = - \rho g dh\]</div>
<p>to be replaced by a constant gravity</p>
<div class="math notranslate nohighlight">
\[dp = - \rho g_{0} dh_{g}\]</div>
<p>providing</p>
<div class="math notranslate nohighlight">
\[dh_{g} = \frac{g}{g_{0}}dh\]</div>
<p>With Newton’s gravitational law this becomes</p>
<div class="math notranslate nohighlight">
\[dh_{g} = \frac{g_{surf}}{g_{0}}\left(\frac{R_{surf}}{R_{surf} + h}\right)^2dh\]</div>
<p>And integrating this, considering that <span class="math notranslate nohighlight">\(h=0\)</span> and <span class="math notranslate nohighlight">\(h_{g}=0\)</span> at the surface, results in</p>
<div class="math notranslate nohighlight">
\[h_{g} = \frac{g_{surf}}{g_{0}}\frac{R_{surf}h}{R_{surf} + h}\]</div>
<div class="math notranslate nohighlight">
\[h = \frac{g_{0}R_{surf}h_{g}}{g_{surf}R_{surf}-g_{0}h_{g}}\]</div>
</div></blockquote>
</div>
<div class="section" id="gas-constant">
<h2>gas constant</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="17%" />
<col width="34%" />
<col width="49%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(k\)</span></td>
<td>Boltzmann constant</td>
<td><span class="math notranslate nohighlight">\(\frac{kg m^2}{K s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(N_A\)</span></td>
<td>Avogadro constant</td>
<td><span class="math notranslate nohighlight">\(\frac{1}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(R\)</span></td>
<td>universal gas constant</td>
<td><span class="math notranslate nohighlight">\(\frac{kg m^2}{K mol s^2}\)</span></td>
</tr>
</tbody>
</table>
<p>Relation between Boltzmann constant, universal gas constant, and Avogadro constant:</p>
<div class="math notranslate nohighlight">
\[k = \frac{R}{N_A}\]</div>
</div></blockquote>
</div>
<div class="section" id="ideal-gas-law">
<h2>ideal gas law</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="14%" />
<col width="35%" />
<col width="51%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(k\)</span></td>
<td>Boltzmann constant</td>
<td><span class="math notranslate nohighlight">\(\frac{kg m^2}{K s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(N\)</span></td>
<td>amount of substance</td>
<td><span class="math notranslate nohighlight">\(molec\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(p\)</span></td>
<td>pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(R\)</span></td>
<td>universal gas constant</td>
<td><span class="math notranslate nohighlight">\(\frac{kg m^2}{K mol s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(T\)</span></td>
<td>temperature</td>
<td><span class="math notranslate nohighlight">\(K\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(V\)</span></td>
<td>volume</td>
<td><span class="math notranslate nohighlight">\(m^3\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\[pV = \frac{NRT}{N_{A}} = NkT\]</div>
</div></blockquote>
</div>
<div class="section" id="barometric-formula">
<h2>barometric formula</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="21%" />
<col width="33%" />
<col width="46%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(g\)</span></td>
<td>gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g_{0}\)</span></td>
<td>mean earth gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(k\)</span></td>
<td>Boltzmann constant</td>
<td><span class="math notranslate nohighlight">\(\frac{kg m^2}{K s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(M_{air}\)</span></td>
<td>molar mass of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(N\)</span></td>
<td>amount of substance</td>
<td><span class="math notranslate nohighlight">\(molec\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(N_A\)</span></td>
<td>Avogadro constant</td>
<td><span class="math notranslate nohighlight">\(\frac{1}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(p\)</span></td>
<td>pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(R\)</span></td>
<td>universal gas constant</td>
<td><span class="math notranslate nohighlight">\(\frac{kg m^2}{K mol s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(T\)</span></td>
<td>temperature</td>
<td><span class="math notranslate nohighlight">\(K\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(V\)</span></td>
<td>volume</td>
<td><span class="math notranslate nohighlight">\(m^3\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(z\)</span></td>
<td>altitude</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(z_{g}\)</span></td>
<td>geopotential height</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\phi\)</span></td>
<td>latitude</td>
<td><span class="math notranslate nohighlight">\(degN\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\rho\)</span></td>
<td>mass density</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{m^3}\)</span></td>
</tr>
</tbody>
</table>
<p>From the ideal gas law we have:</p>
<div class="math notranslate nohighlight">
\[p = \frac{NkT}{V} = \frac{10^{-3}NM_{air}}{VN_{a}}\frac{kTN_{a}}{10^{-3}M_{air}} = \rho\frac{RT}{10^{-3}M_{air}}\]</div>
<p>And from the hydrostatic assumption we get:</p>
<div class="math notranslate nohighlight">
\[dp = - \rho g dz\]</div>
<p>Dividing <span class="math notranslate nohighlight">\(dp\)</span> by <cite>p</cite> we get:</p>
<div class="math notranslate nohighlight">
\[\frac{dp}{p} = -\frac{10^{-3}M_{air}\rho g dz}{\rho RT} = -\frac{10^{-3}M_{air}gdz}{RT}\]</div>
<p>Integrating this expression from one pressure level to the next we get:</p>
<div class="math notranslate nohighlight">
\[p(i+1) = p(i)e^{-\int^{z(i+1)}_{z(i)}\frac{10^{-3}M_{air}g}{RT}dz}\]</div>
<p>We can approximate this further by using an average value of the height dependent quantities
<span class="math notranslate nohighlight">\(M_{air}\)</span>, <span class="math notranslate nohighlight">\(g\)</span> and <span class="math notranslate nohighlight">\(T\)</span> for the integration over the range <span class="math notranslate nohighlight">\([z(i),z(i+1)]\)</span>.
This gives:</p>
<div class="math notranslate nohighlight">
\begin{eqnarray}
g & = & g(\phi,\frac{z(i)+z(i+1)}{2}) \\
p(i+1) & = & p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{2}\frac{2}{T(i)+T(i+1)}\frac{g}{R}\left(z(i+1)-z(i)\right)} \\
& = & p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{T(i)+T(i+1)}\frac{g}{R}\left(z(i+1)-z(i)\right)}
\end{eqnarray}</div><p>When using geopotential height the formula is the same except that <span class="math notranslate nohighlight">\(g=g_{0}\)</span> at all levels:</p>
<div class="math notranslate nohighlight">
\[p(i+1) = p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{T(i)+T(i+1)}\frac{g_{0}}{R}\left(z_{g}(i+1)-z_{g}(i)\right)}\]</div>
</div></blockquote>
</div>
<div class="section" id="mass-density">
<h2>mass density</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="25%" />
<col width="38%" />
<col width="37%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(N\)</span></td>
<td>amount of substance</td>
<td><span class="math notranslate nohighlight">\(molec\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(N_A\)</span></td>
<td>Avogadro constant</td>
<td><span class="math notranslate nohighlight">\(\frac{1}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(M_{air}\)</span></td>
<td>molar mass of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(V\)</span></td>
<td>volume</td>
<td><span class="math notranslate nohighlight">\(m^3\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\rho\)</span></td>
<td>mass density</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{m^3}\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\[\rho = \frac{10^{-3}NM_{air}}{VN_{a}}\]</div>
</div></blockquote>
</div>
<div class="section" id="number-density">
<h2>number density</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="17%" />
<col width="36%" />
<col width="47%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(n\)</span></td>
<td>number density</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(N\)</span></td>
<td>amount of substance</td>
<td><span class="math notranslate nohighlight">\(molec\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(V\)</span></td>
<td>volume</td>
<td><span class="math notranslate nohighlight">\(m^3\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\[n = \frac{N}{V}\]</div>
</div></blockquote>
</div>
<div class="section" id="dry-air-vs-total-air">
<h2>dry air vs. total air</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="31%" />
<col width="36%" />
<col width="33%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(n\)</span></td>
<td>number density of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(n_{dry\_air}\)</span></td>
<td>number density of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(n_{H_{2}O}\)</span></td>
<td>number density of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(M_{air}\)</span></td>
<td>molar mass of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(M_{dry\_air}\)</span></td>
<td>molar mass of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(M_{H_{2}O}\)</span></td>
<td>molar mass of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\rho\)</span></td>
<td>mass density of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\rho_{dry\_air}\)</span></td>
<td>mass density of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\rho_{H_{2}O}\)</span></td>
<td>mass density of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{m^3}\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\begin{eqnarray}
n & = & n_{dry\_air} + n_{H_{2}O} \\
M_{air}n & = & M_{dry\_air}n_{dry\_air} + M_{H_{2}O}n_{H_{2}O} \\
\rho & = & \rho_{dry\_air} + \rho_{H_{2}O} \\
\end{eqnarray}</div></div></blockquote>
</div>
<div class="section" id="virtual-temperature">
<h2>virtual temperature</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="26%" />
<col width="32%" />
<col width="42%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(k\)</span></td>
<td>Boltzmann constant</td>
<td><span class="math notranslate nohighlight">\(\frac{kg m^2}{K s^2}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(M_{air}\)</span></td>
<td>molar mass of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(M_{dry\_air}\)</span></td>
<td>molar mass of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(M_{H_{2}O}\)</span></td>
<td>molar mass of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(N\)</span></td>
<td>amount of substance</td>
<td><span class="math notranslate nohighlight">\(molec\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(N_A\)</span></td>
<td>Avogadro constant</td>
<td><span class="math notranslate nohighlight">\(\frac{1}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(p\)</span></td>
<td>pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(p_{dry\_air}\)</span></td>
<td>dry air partial pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(p_{H_{2}O}\)</span></td>
<td>H2O partial pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(R\)</span></td>
<td>universal gas constant</td>
<td><span class="math notranslate nohighlight">\(\frac{kg m^2}{K mol s^2}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(T\)</span></td>
<td>temperature</td>
<td><span class="math notranslate nohighlight">\(K\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(T_{v}\)</span></td>
<td>virtual temperature</td>
<td><span class="math notranslate nohighlight">\(K\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(V\)</span></td>
<td>volume</td>
<td><span class="math notranslate nohighlight">\(m^3\)</span></td>
</tr>
</tbody>
</table>
<p>From the ideal gas law we have:</p>
<div class="math notranslate nohighlight">
\[p = \frac{NkT}{V} = \frac{10^{-3}NM_{air}}{VN_{a}}\frac{kTN_{a}}{10^{-3}M_{air}} = \rho \frac{RT}{10^{-3}M_{air}}\]</div>
<p>The virtual temperature allows us to use the dry air molar mass in this equation:</p>
<div class="math notranslate nohighlight">
\[p = \rho\frac{RT_{v}}{10^{-3}M_{dry\_air}}\]</div>
<p>This gives:</p>
<div class="math notranslate nohighlight">
\[T_{v} = \frac{M_{dry\_air}}{M_{air}}T\]</div>
</div></blockquote>
</div>
<div class="section" id="volume-mixing-ratio">
<h2>volume mixing ratio</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="28%" />
<col width="40%" />
<col width="32%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(n\)</span></td>
<td>number density of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(n_{dry\_air}\)</span></td>
<td>number density of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(n_{H_{2}O}\)</span></td>
<td>number density of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(n_{x}\)</span></td>
<td>number density of quantity x</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\nu_{x}\)</span></td>
<td>volume mixing ratio of quantity
x with regard to total air</td>
<td><span class="math notranslate nohighlight">\(ppv\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\bar{\nu}_{x}\)</span></td>
<td>volume mixing ratio of quantity
x with regard to dry air</td>
<td><span class="math notranslate nohighlight">\(ppv\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\begin{eqnarray}
\nu_{x} & = & \frac{n_{x}}{n} \\
\bar{\nu}_{x} & = & \frac{n_{x}}{n_{dry\_air}} \\
\nu_{dry\_air} & = & \frac{n_{dry\_air}}{n} =
\frac{n - n_{H_{2}O}}{n} = 1 - \nu_{H_{2}O} \\
\nu_{air} & = & \frac{n}{n} = 1 \\
\bar{\nu}_{dry\_air} & = & \frac{n_{dry\_air}}{n_{dry\_air}} = 1 \\
\bar{\nu}_{H_{2}O} & = & \frac{n_{H_{2}O}}{n_{dry\_air}} =
\frac{\nu_{H_{2}O}}{\nu_{dry\_air}} = \frac{\nu_{H_{2}O}}{1 - \nu_{H_{2}O}} \\
\nu_{H_{2}O} & = & \frac{\bar{\nu}_{H_{2}O}}{1 + \bar{\nu}_{H_{2}O}}
\end{eqnarray}</div></div></blockquote>
</div>
<div class="section" id="mass-mixing-ratio">
<h2>mass mixing ratio</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="27%" />
<col width="40%" />
<col width="32%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(M_{air}\)</span></td>
<td>molar mass of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(M_{dry\_air}\)</span></td>
<td>molar mass of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(M_{x}\)</span></td>
<td>molar mass of quantity x</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(n\)</span></td>
<td>number density of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(n_{dry\_air}\)</span></td>
<td>number density of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(n_{H_{2}O}\)</span></td>
<td>number density of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(n_{x}\)</span></td>
<td>number density of quantity x</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(q_{x}\)</span></td>
<td>mass mixing ratio of quantity x
with regard to total air</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{kg}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\bar{q}_{x}\)</span></td>
<td>mass mixing ratio of quantity x
with regard to dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{kg}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\nu_{x}\)</span></td>
<td>volume mixing ratio of quantity
x with regard to total air</td>
<td><span class="math notranslate nohighlight">\(ppv\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\bar{\nu}_{x}\)</span></td>
<td>volume mixing ratio of quantity
x with regard to dry air</td>
<td><span class="math notranslate nohighlight">\(ppv\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\begin{eqnarray}
q_{x} & = & \frac{n_{x}M_{x}}{nM_{air}} = \nu_{x}\frac{M_{x}}{M_{air}} \\
\bar{q}_{x} & = & \frac{n_{x}M_{x}}{n_{dry\_air}M_{dry\_air}} = \bar{\nu}_{x}\frac{M_{x}}{M_{dry\_air}} \\
q_{dry\_air} & = & \frac{n_{dry\_air}M_{dry\_air}}{nM_{air}} =
\frac{nM_{air} - n_{H_{2}O}M_{H_{2}O}}{nM_{air}} = 1 - q_{H_{2}O} \\
q_{air} & = & \frac{nM_{air}}{nM_{air}} = 1 \\
\bar{q}_{dry\_air} & = & \frac{n_{dry\_air}M_{dry\_air}}{n_{dry\_air}M_{dry\_air}} = 1 \\
\bar{q}_{H_{2}O} & = & \frac{n_{H_{2}O}M_{H_{2}O}}{n_{dry\_air}M_{dry\_air}} =
\frac{q_{H_{2}O}}{q_{dry\_air}} = \frac{q_{H_{2}O}}{1 - q_{H_{2}O}} \\
q_{H_{2}O} & = & \frac{\bar{q}_{H_{2}O}}{1 + \bar{q}_{H_{2}O}}
\end{eqnarray}</div></div></blockquote>
</div>
<div class="section" id="molar-mass-of-total-air">
<h2>molar mass of total air</h2>
<ol class="arabic">
<li><p class="first">molar mass of total air from H2O volume mixing ratio</p>
<table border="1" class="docutils">
<colgroup>
<col width="28%" />
<col width="38%" />
<col width="35%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(M_{air}\)</span></td>
<td>molar mass of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(M_{dry\_air}\)</span></td>
<td>molar mass of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(M_{H_{2}O}\)</span></td>
<td>molar mass of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(n\)</span></td>
<td>number density of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(n_{dry\_air}\)</span></td>
<td>number density of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(n_{H_{2}O}\)</span></td>
<td>number density of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\nu_{H_{2}O}\)</span></td>
<td>volume mixing ratio of H2O</td>
<td><span class="math notranslate nohighlight">\(ppv\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\begin{eqnarray}
M_{air} & = & \frac{M_{dry\_air}n_{dry\_air} + M_{H_{2}O}n_{H_{2}O}}{n} \\
& = & M_{dry\_air}\left(1 - \nu_{H_{2}O}\right) + M_{H_{2}O}\nu_{H_{2}O}
\end{eqnarray}</div></li>
<li><p class="first">molar mass of total air from H2O mass mixing ratio</p>
<table border="1" class="docutils">
<colgroup>
<col width="28%" />
<col width="38%" />
<col width="35%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(M_{air}\)</span></td>
<td>molar mass of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(M_{dry\_air}\)</span></td>
<td>molar mass of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(M_{H_{2}O}\)</span></td>
<td>molar mass of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(n\)</span></td>
<td>number density of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(n_{dry\_air}\)</span></td>
<td>number density of dry air</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(n_{H_{2}O}\)</span></td>
<td>number density of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{molec}{m^3}\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(q_{H_{2}O}\)</span></td>
<td>mass mixing ratio of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{kg}\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\nu_{H_{2}O}\)</span></td>
<td>volume mixing ratio of H2O</td>
<td><span class="math notranslate nohighlight">\(\frac{kg}{kg}\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\begin{eqnarray}
M_{air} & = & M_{dry\_air}\left(1 - \nu_{H_{2}O}\right) + M_{H_{2}O}\nu_{H_{2}O} \\
& = & M_{dry\_air}\left(1 - \frac{M_{air}}{M_{H_{2}O}}q_{H_{2}O}\right) + M_{air}q_{H_{2}O} \\
& = & \frac{M_{dry\_air}}{1 + \frac{M_{dry\_air}}{M_{H_{2}O}}q_{H_{2}O} - q_{H_{2}O}} \\
& = & \frac{M_{H_{2}O}M_{dry\_air}}{M_{H_{2}O} + M_{dry\_air}q_{H_{2}O} - M_{H_{2}O}q_{H_{2}O}} \\
& = & \frac{M_{H_{2}O}M_{dry\_air}}{\left(1-q_{H_{2}O}\right)M_{H_{2}O} + q_{H_{2}O}M_{dry\_air}} \\
\end{eqnarray}</div></li>
</ol>
</div>
<div class="section" id="partial-pressure">
<h2>partial pressure</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="33%" />
<col width="49%" />
<col width="17%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(p\)</span></td>
<td>total pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(p_{x}\)</span></td>
<td>partial pressure of quantity</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\nu_{x}\)</span></td>
<td>volume mixing ratio of quantity
x with regard to total air</td>
<td><span class="math notranslate nohighlight">\(ppv\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\bar{\nu}_{x}\)</span></td>
<td>volume mixing ratio of quantity
x with regard to dry air</td>
<td><span class="math notranslate nohighlight">\(ppv\)</span></td>
</tr>
</tbody>
</table>
<div class="math notranslate nohighlight">
\begin{eqnarray}
p_{x} & = & \nu_{x}p \\
p_{x} & = & \bar{\nu}_{x}p_{dry\_air} \\
p_{x} & = & N_{x}kT
\end{eqnarray}</div></div></blockquote>
</div>
<div class="section" id="saturated-water-vapor-pressure">
<h2>saturated water vapor pressure</h2>
<blockquote>
<div><table border="1" class="docutils">
<colgroup>
<col width="24%" />
<col width="56%" />
<col width="20%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">name</th>
<th class="head">unit</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(e_{w}\)</span></td>
<td>saturated water vapor pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(T\)</span></td>
<td>temperature</td>
<td><span class="math notranslate nohighlight">\(K\)</span></td>
</tr>
</tbody>
</table>
<p>This is the August-Roche-Magnus formula for the saturated water vapour pressure</p>
<div class="math notranslate nohighlight">
\[e_{w} = 610.94e^{\frac{17.625(T-273.15)}{(T-273.15)+243.04}}\]</div>
</div></blockquote>
</div>
</div>
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