# temperature derivations

1. temperature from pressure and number density

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.

$T = \frac{p}{kn}$
2. temperature from virtual temperature

symbol

description

unit

variable name

$$M_{air}$$

molar mass of total air

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

molar_mass {:}

$$M_{dry\_air}$$

molar mass of dry air

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

$$T$$

temperature

$$K$$

temperature {:}

$$T_{v}$$

virtual temperature

$$K$$

virtual_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.

$T = \frac{M_{air}}{M_{dry\_air}}T_{v}$
3. surface temperature from surface pressure and surface number density

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.

$T_{surf} = \frac{p_{surf}}{kn_{surf}}$