Common formula

gravity

1. 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	\(m\)

   \[g = g_{surf}\left(\frac{R_{surf}}{R_{surf} + h}\right)^2\]

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}

3. 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

1. 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}

2. 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

\[e_{w} = 610.94e^{\frac{17.625(T-273.15)}{(T-273.15)+243.04}}\]