Common formula
gravity
Newton’s gravitational law
symbol description unit \(g\) gravity \(\frac{m}{s^2}\) \(g_{surf}\) gravity at earth surface \(\frac{m}{s^2}\) \(h\) height above surface \(m\) \(R_{surf}\) earth radius at surface \(m\) \[g = g_{surf}\left(\frac{R_{surf}}{R_{surf} + h}\right)^2\]normal gravity at ellipsoid surface
This is the WGS84 ellipsoidal gravity formula as taken from NIMA TR8350.2
symbol name unit \(a\) WGS84 semi-major axis \(m\) \(b\) WGS84 semi-minor axis \(m\) \(e\) eccentricity \(m\) \(g_{e}\) gravity at equator \(\frac{m}{s^2}\) \(g_{p}\) gravity at poles \(\frac{m}{s^2}\) \(g_{surf}\) gravity at surface \(\frac{m}{s^2}\) \(\phi\) latitude \(degN\) \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}normal gravity above ellipsoid surface
This is the WGS84 ellipsoidal gravity formula as taken from NIMA TR8350.2
symbol name unit \(a\) WGS84 semi-major axis \(m\) \(b\) WGS84 semi-minor axis \(m\) \(f\) WGS84 flattening \(m\) \(g\) gravity \(\frac{m}{s^2}\) \(g_{surf}\) gravity at the ellipsoid surface \(\frac{m}{s^2}\) \(GM\) WGS84 earth’s gravitational constant \(\frac{m^3}{s^2}\) \(z\) altitude \(m\) \(\phi\) latitude \(degN\) \(\omega\) WGS84 earth angular velocity \(rad/s\) The formula used is the one based on the truncated Taylor series expansion:
\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}
geopotential height
symbol description unit \(g\) gravity \(\frac{m}{s^2}\) \(g_{0}\) mean earth gravity \(\frac{m}{s^2}\) \(g_{surf}\) gravity at earth surface \(\frac{m}{s^2}\) \(p\) pressure \(Pa\) \(R_{surf}\) earth radius at surface \(m\) \(h\) height above surface \(m\) \(h_{g}\) geopotential height \(m\) \(\phi\) latitude \(degN\) \(\rho\) mass density \(\frac{kg}{m^3}\) The geopotential height allows the gravity in the hydrostatic equation
\[dp = - \rho g dh\]to be replaced by a constant gravity
\[dp = - \rho g_{0} dh_{g}\]providing
\[dh_{g} = \frac{g}{g_{0}}dh\]With Newton’s gravitational law this becomes
\[dh_{g} = \frac{g_{surf}}{g_{0}}\left(\frac{R_{surf}}{R_{surf} + h}\right)^2dh\]And integrating this, considering that \(h=0\) and \(h_{g}=0\) at the surface, results in
\[h_{g} = \frac{g_{surf}}{g_{0}}\frac{R_{surf}h}{R_{surf} + h}\]\[h = \frac{g_{0}R_{surf}h_{g}}{g_{surf}R_{surf}-g_{0}h_{g}}\]
gas constant
symbol name unit \(k\) Boltzmann constant \(\frac{kg m^2}{K s^2}\) \(N_A\) Avogadro constant \(\frac{1}{mol}\) \(R\) universal gas constant \(\frac{kg m^2}{K mol s^2}\) Relation between Boltzmann constant, universal gas constant, and Avogadro constant:
\[k = \frac{R}{N_A}\]
ideal gas law
symbol name unit \(k\) Boltzmann constant \(\frac{kg m^2}{K s^2}\) \(N\) amount of substance \(molec\) \(p\) pressure \(Pa\) \(R\) universal gas constant \(\frac{kg m^2}{K mol s^2}\) \(T\) temperature \(K\) \(V\) volume \(m^3\) \[pV = \frac{NRT}{N_{A}} = NkT\]
barometric formula
symbol name unit \(g\) gravity \(\frac{m}{s^2}\) \(g_{0}\) mean earth gravity \(\frac{m}{s^2}\) \(k\) Boltzmann constant \(\frac{kg m^2}{K s^2}\) \(M_{air}\) molar mass of total air \(\frac{g}{mol}\) \(N\) amount of substance \(molec\) \(N_A\) Avogadro constant \(\frac{1}{mol}\) \(p\) pressure \(Pa\) \(R\) universal gas constant \(\frac{kg m^2}{K mol s^2}\) \(T\) temperature \(K\) \(V\) volume \(m^3\) \(z\) altitude \(m\) \(z_{g}\) geopotential height \(m\) \(\phi\) latitude \(degN\) \(\rho\) mass density \(\frac{kg}{m^3}\) From the ideal gas law we have:
\[p = \frac{NkT}{V} = \frac{10^{-3}NM_{air}}{VN_{a}}\frac{kTN_{a}}{10^{-3}M_{air}} = \rho\frac{RT}{10^{-3}M_{air}}\]And from the hydrostatic assumption we get:
\[dp = - \rho g dz\]Dividing \(dp\) by p we get:
\[\frac{dp}{p} = -\frac{10^{-3}M_{air}\rho g dz}{\rho RT} = -\frac{10^{-3}M_{air}gdz}{RT}\]Integrating this expression from one pressure level to the next we get:
\[p(i+1) = p(i)e^{-\int^{z(i+1)}_{z(i)}\frac{10^{-3}M_{air}g}{RT}dz}\]We can approximate this further by using an average value of the height dependent quantities \(M_{air}\), \(g\) and \(T\) for the integration over the range \([z(i),z(i+1)]\). This gives:
\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}When using geopotential height the formula is the same except that \(g=g_{0}\) at all levels:
\[p(i+1) = p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{T(i)+T(i+1)}\frac{g_{0}}{R}\left(z_{g}(i+1)-z_{g}(i)\right)}\]
mass density
symbol name unit \(N\) amount of substance \(molec\) \(N_A\) Avogadro constant \(\frac{1}{mol}\) \(M_{air}\) molar mass of total air \(\frac{g}{mol}\) \(V\) volume \(m^3\) \(\rho\) mass density \(\frac{kg}{m^3}\) \[\rho = \frac{10^{-3}NM_{air}}{VN_{a}}\]
number density
symbol name unit \(n\) number density \(\frac{molec}{m^3}\) \(N\) amount of substance \(molec\) \(V\) volume \(m^3\) \[n = \frac{N}{V}\]
dry air vs. total air
symbol name unit \(n\) number density of total air \(\frac{molec}{m^3}\) \(n_{dry\_air}\) number density of dry air \(\frac{molec}{m^3}\) \(n_{H_{2}O}\) number density of H2O \(\frac{molec}{m^3}\) \(M_{air}\) molar mass of total air \(\frac{g}{mol}\) \(M_{dry\_air}\) molar mass of dry air \(\frac{g}{mol}\) \(M_{H_{2}O}\) molar mass of H2O \(\frac{g}{mol}\) \(\rho\) mass density of total air \(\frac{kg}{m^3}\) \(\rho_{dry\_air}\) mass density of dry air \(\frac{kg}{m^3}\) \(\rho_{H_{2}O}\) mass density of H2O \(\frac{kg}{m^3}\) \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}
virtual temperature
symbol name unit \(k\) Boltzmann constant \(\frac{kg m^2}{K s^2}\) \(M_{air}\) molar mass of total air \(\frac{g}{mol}\) \(M_{dry\_air}\) molar mass of dry air \(\frac{g}{mol}\) \(M_{H_{2}O}\) molar mass of H2O \(\frac{g}{mol}\) \(N\) amount of substance \(molec\) \(N_A\) Avogadro constant \(\frac{1}{mol}\) \(p\) pressure \(Pa\) \(p_{dry\_air}\) dry air partial pressure \(Pa\) \(p_{H_{2}O}\) H2O partial pressure \(Pa\) \(R\) universal gas constant \(\frac{kg m^2}{K mol s^2}\) \(T\) temperature \(K\) \(T_{v}\) virtual temperature \(K\) \(V\) volume \(m^3\) From the ideal gas law we have:
\[p = \frac{NkT}{V} = \frac{10^{-3}NM_{air}}{VN_{a}}\frac{kTN_{a}}{10^{-3}M_{air}} = \rho \frac{RT}{10^{-3}M_{air}}\]The virtual temperature allows us to use the dry air molar mass in this equation:
\[p = \rho\frac{RT_{v}}{10^{-3}M_{dry\_air}}\]This gives:
\[T_{v} = \frac{M_{dry\_air}}{M_{air}}T\]
volume mixing ratio
symbol name unit \(n\) number density of total air \(\frac{molec}{m^3}\) \(n_{dry\_air}\) number density of dry air \(\frac{molec}{m^3}\) \(n_{H_{2}O}\) number density of H2O \(\frac{molec}{m^3}\) \(n_{x}\) number density of quantity x \(\frac{molec}{m^3}\) \(\nu_{x}\) volume mixing ratio of quantity x with regard to total air \(ppv\) \(\bar{\nu}_{x}\) volume mixing ratio of quantity x with regard to dry air \(ppv\) \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}
mass mixing ratio
symbol name unit \(M_{air}\) molar mass of total air \(\frac{g}{mol}\) \(M_{dry\_air}\) molar mass of dry air \(\frac{g}{mol}\) \(M_{x}\) molar mass of quantity x \(\frac{g}{mol}\) \(n\) number density of total air \(\frac{molec}{m^3}\) \(n_{dry\_air}\) number density of dry air \(\frac{molec}{m^3}\) \(n_{H_{2}O}\) number density of H2O \(\frac{molec}{m^3}\) \(n_{x}\) number density of quantity x \(\frac{molec}{m^3}\) \(q_{x}\) mass mixing ratio of quantity x with regard to total air \(\frac{kg}{kg}\) \(\bar{q}_{x}\) mass mixing ratio of quantity x with regard to dry air \(\frac{kg}{kg}\) \(\nu_{x}\) volume mixing ratio of quantity x with regard to total air \(ppv\) \(\bar{\nu}_{x}\) volume mixing ratio of quantity x with regard to dry air \(ppv\) \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}
molar mass of total air
molar mass of total air from H2O volume mixing ratio
symbol name unit \(M_{air}\) molar mass of total air \(\frac{g}{mol}\) \(M_{dry\_air}\) molar mass of dry air \(\frac{g}{mol}\) \(M_{H_{2}O}\) molar mass of H2O \(\frac{g}{mol}\) \(n\) number density of total air \(\frac{molec}{m^3}\) \(n_{dry\_air}\) number density of dry air \(\frac{molec}{m^3}\) \(n_{H_{2}O}\) number density of H2O \(\frac{molec}{m^3}\) \(\nu_{H_{2}O}\) volume mixing ratio of H2O \(ppv\) \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}molar mass of total air from H2O mass mixing ratio
symbol name unit \(M_{air}\) molar mass of total air \(\frac{g}{mol}\) \(M_{dry\_air}\) molar mass of dry air \(\frac{g}{mol}\) \(M_{H_{2}O}\) molar mass of H2O \(\frac{g}{mol}\) \(n\) number density of total air \(\frac{molec}{m^3}\) \(n_{dry\_air}\) number density of dry air \(\frac{molec}{m^3}\) \(n_{H_{2}O}\) number density of H2O \(\frac{molec}{m^3}\) \(q_{H_{2}O}\) mass mixing ratio of H2O \(\frac{kg}{kg}\) \(\nu_{H_{2}O}\) volume mixing ratio of H2O \(\frac{kg}{kg}\) \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}
partial pressure
symbol name unit \(p\) total pressure \(Pa\) \(p_{x}\) partial pressure of quantity \(Pa\) \(\nu_{x}\) volume mixing ratio of quantity x with regard to total air \(ppv\) \(\bar{\nu}_{x}\) volume mixing ratio of quantity x with regard to dry air \(ppv\) \begin{eqnarray} p_{x} & = & \nu_{x}p \\ p_{x} & = & \bar{\nu}_{x}p_{dry\_air} \\ p_{x} & = & N_{x}kT \end{eqnarray}
saturated water vapor pressure
symbol name unit \(e_{w}\) saturated water vapor pressure \(Pa\) \(T\) temperature \(K\) This is the August-Roche-Magnus formula for the saturated water vapour pressure
\[e_{w} = 610.94e^{\frac{17.625(T-273.15)}{(T-273.15)+243.04}}\]