# pressure derivations

1. pressure from bounds

symbol

description

unit

variable name

$$p$$

pressure

$$Pa$$

pressure {:}

$$p^{B}(l)$$

pressure boundaries ($$l \in \{1,2\}$$)

$$Pa$$

pressure_bounds {:,2}

The pattern : for the dimensions can represent {vertical}, or {time,vertical}.

$p = e^{\frac{ln(z^{B}(2)) + ln(z^{B}(1))}{2}}$
2. pressure from altitude

symbol

description

unit

variable name

$$a$$

WGS84 semi-major axis

$$m$$

$$b$$

WGS84 semi-minor axis

$$m$$

$$f$$

WGS84 flattening

$$m$$

$$g$$

normal gravity at sea level

$$\frac{m}{s^2}$$

$$g_{0}$$

mean earth gravity

$$\frac{m}{s^2}$$

$$g_{h}$$

gravity at specific height

$$\frac{m}{s^2}$$

$$GM$$

WGS84 earth’s gravitational constant

$$\frac{m^3}{s^2}$$

$$M_{air}(i)$$

molar mass of total air

$$\frac{g}{mol}$$

molar_mass {:,vertical}

$$p(i)$$

pressure

$$Pa$$

pressure {:,vertical}

$$p_{surf}$$

surface pressure

$$Pa$$

surface_pressure {:}

$$R$$

universal gas constant

$$\frac{kg m^2}{K mol s^2}$$

$$T(i)$$

temperature

$$K$$

temperature {:,vertical}

$$z(i)$$

altitude

$$m$$

altitude {:,vertical}

$$z_{surf}$$

surface height

$$m$$

surface_altitude {:}

$$\phi$$

latitude

$$degN$$

latitude {:}

$$\omega$$

WGS84 earth angular velocity

$$rad/s$$

The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

The surface pressure $$p_{surf}$$ and surface height $$z_{surf}$$ need to use the same definition of ‘surface’.

The altitudes $$z(i)$$ are expected to be above the surface height. This should normally be the case since even for altitude grids that start at the surface, $$z_{surf}$$ should equal the lower altitude boundary $$z^{B}(1,1)$$, whereas $$z(1)$$ should then be between $$z^{B}(1,1)$$ and $$z^{B}(1,2)$$ (which is generally not equal to $$z^{B}(1,1)$$).

\begin{eqnarray} g & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013 {\sin}^2(\frac{\pi}{180}\phi)}} \\ m & = & \frac{\omega^2a^2b}{GM} \\ g(1) & = & g \left(1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)\frac{z_{surf}+z(1)}{2} + \frac{3}{a^2}\left(\frac{z_{surf}+z(1)}{2}\right)^2\right) \\ g(i) & = & g \left(1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)\frac{z(i-1)+z(i)}{2} + \frac{3}{a^2}\left(\frac{z(i-1)+z(i)}{2}\right)^2\right), 1 < i \leq N \\ p(1) & = & p_{surf}e^{-10^{-3}\frac{M_{air}(1)}{T(1)}\frac{g_{h}(1)}{R}\left(z(i)-z_{surf}\right)} \\ p(i) & = & p(i-1)e^{-10^{-3}\frac{M_{air}(i-1)+M_{air}(i)}{T(i-1)+T(i)}\frac{g_{h}(i)}{R}\left(z(i)-z(i-1)\right)}, 1 < i \leq N \end{eqnarray}
3. pressure from geopotential height

symbol

description

unit

variable name

$$g_{0}$$

mean earth gravity

$$\frac{m}{s^2}$$

$$M_{air}(i)$$

molar mass of total air

$$\frac{g}{mol}$$

molar_mass {:,vertical}

$$p(i)$$

pressure

$$Pa$$

pressure {:,vertical}

$$p_{surf}$$

surface pressure

$$Pa$$

surface_pressure {:}

$$R$$

universal gas constant

$$\frac{kg m^2}{K mol s^2}$$

$$T(i)$$

temperature

$$K$$

temperature {:,vertical}

$$z_{g}(i)$$

geopotential height

$$m$$

geopotential_height {:,vertical}

$$z_{g,surf}$$

surface geopotential height

$$m$$

surface_geopotential_height {:}

The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

The surface pressure $$p_{surf}$$ and surface height $$z_{g,surf}$$ need to use the same definition of ‘surface’.

The geopotential heights $$z_{g}(i)$$ are expected to be above the surface geopotential height. This should normally be the case since even for geopotential height grids that start at the surface, $$z_{g,surf}$$ should equal the lower altitude boundary $$z^{B}_{g}(1,1)$$, whereas $$z_{g}(1)$$ should then be between $$z^{B}_{g}(1,1)$$ and $$z^{B}_{g}(1,2)$$ (which is generally not equal to $$z^{B}_{g}(1,1)$$).

\begin{eqnarray} p(1) & = & p_{surf}e^{-10^{-3}\frac{M_{air}(1)}{T(1)}\frac{g_{0}}{R}\left(z_{g}(i)-z_{g,surf}\right)} \\ p(i) & = & p(i-1)e^{-10^{-3}\frac{M_{air}(i-1)+M_{air}(i)}{T(i-1)+T(i)}\frac{g_{0}}{R}\left(z_{g}(i)-z_{g}(i-1)\right)}, 1 < i \leq N \end{eqnarray}
4. pressure from number density and temperature

symbol

description

unit

variable name

$$k$$

Boltzmann constant

$$\frac{kg m^2}{K s^2}$$

$$n$$

number density

$$\frac{molec}{m^3}$$

number_density {:}

$$p$$

pressure

$$Pa$$

pressure {:}

$$T$$

temperature

$$K$$

temperature {:}

The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

$p = nkT$
5. surface pressure from surface number density and surface temperature

symbol

description

unit

variable name

$$k$$

Boltzmann constant

$$\frac{kg m^2}{K s^2}$$

$$n_{surf}$$

surface number density

$$\frac{molec}{m^3}$$

surface_number_density {:}

$$p_{surf}$$

surface pressure

$$Pa$$

surface_pressure {:}

$$T_{surf}$$

surface temperature

$$K$$

surface_temperature {:}

The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

$p_{surf} = n_{surf}kT_{surf}$
6. tropopause pressure from temperature and altitude/pressure

symbol

description

unit

variable name

$$p(i)$$

pressure

$$Pa$$

pressure {:,vertical}

$$p_{TP}$$

tropopause pressure

$$Pa$$

tropopause_pressure {:}

$$T(i)$$

temperature

$$K$$

temperature {:,vertical}

$$z(i)$$

altitude

$$m$$

altitude {:,vertical}

The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

The tropopause pressure $$p_{TP}$$ equals the pressure $$p(i)$$ where $$i$$ is the minimum level that satisfies:

\begin{eqnarray} & 1 < i < N & \wedge \\ & 5000 <= p(i) <= 50000 & \wedge \\ & \frac{T(i-1)-T(i)}{z(i)-z(i-1)} > 0.002 \wedge \frac{T(i)-T(i+1)}{z(i+1)-z(i)} <= 0.002 & \wedge \\ & \frac{\sum_{j, i < j < N \wedge z(j+1)-z(i) <= 2000} \frac{T(j)-T(j+1)}{z(j+1)-z(j)}}{\sum_{j, i < j < N \wedge z(j+1)-z(i) <= 2000}{1}} <= 0.002 & \end{eqnarray}

If no such $$i$$ can be found then $$p_{TP}$$ is set to NaN.