# geopotential height derivations

1. geopotential height from geopotential

symbol

description

unit

variable name

$$g_{0}$$

mean earth gravity

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

$$z_{g}$$

geopotential height

$$m$$

geopotential_height {:}

$$\Phi$$

geopotential

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

geopotential {:}

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.

$z_{g} = \frac{\Phi}{g_{0}}$
2. geopotential height from altitude

symbol

description

unit

variable name

$$g_{0}$$

mean earth gravity

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

$$g$$

nominal gravity at sea level

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

$$R$$

$$m$$

$$z$$

altitude

$$m$$

altitude {:}

$$z_{g}$$

geopotential height

$$m$$

geopotential_height {:}

$$\phi$$

latitude

$$degN$$

latitude {:}

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

\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)}} \\ R & = & \frac{1}{\sqrt{\left(\frac{\cos(\frac{\pi}{180}\phi)}{6356752.0}\right)^2 + \left(\frac{\sin(\frac{\pi}{180}\phi)}{6378137.0}\right)^2}} \\ z_{g} & = & \frac{g}{g_{0}}\frac{Rz}{z + R} \end{eqnarray}
3. geopotential height from pressure

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 pressures $$p(i)$$ are expected to be at higher levels than the surface pressure (i.e. lower values). This should normally be the case since even for pressure grids that start at the surface, $$p_{surf}$$ should equal the lower pressure boundary $$p^{B}(1,1)$$, whereas $$p(1)$$ should then be between $$p^{B}(1,1)$$ and $$p^{B}(1,2)$$ (which is generally not equal to $$p^{B}(1,1)$$).

\begin{eqnarray} z_{g}(1) & = & z_{g,surf} + 10^{3}\frac{T(1)}{M_{air}(1)}\frac{R}{g_{0}}\ln\left(\frac{p_{surf}}{p(i)}\right) \\ z_{g}(i) & = & z_{g}(i-1) + 10^{3}\frac{T(i-1)+T(i)}{M_{air}(i-1)+M_{air}(i)}\frac{R}{g_{0}}\ln\left(\frac{p(i-1)}{p(i)}\right), 1 < i \leq N \end{eqnarray}
4. surface geopotential height from surface geopotential

symbol

description

unit

variable name

$$g_{0}$$

mean earth gravity

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

$$z_{g,surf}$$

surface geopotential height

$$m$$

surface_geopotential_height {:}

$$\Phi_{surf}$$

surface geopotential

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

surface_geopotential {:}

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

$z_{g,surf} = \frac{\Phi_{surf}}{g_{0}}$
5. surface geopotential height from surface altitude

symbol

description

unit

variable name

$$g_{0}$$

mean earth gravity

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

$$g$$

nominal gravity at sea level

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

$$R$$

$$m$$

$$z_{surf}$$

surface altitude

$$m$$

surface_altitude {:}

$$z_{g,surf}$$

surface geopotential height

$$m$$

surface_geopotential_height {:}

$$\phi$$

latitude

$$degN$$

latitude {:}

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

\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)}} \\ R & = & \frac{1}{\sqrt{\left(\frac{\cos(\frac{\pi}{180}\phi)}{6356752.0}\right)^2 + \left(\frac{\sin(\frac{\pi}{180}\phi)}{6378137.0}\right)^2}} \\ z_{g,surf} & = & \frac{g}{g_{0}}\frac{Rz_{surf}}{z_{surf} + R} \end{eqnarray}