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 {:}
   $\mathrm{\Phi }$	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.

   $$z_g=\frac{\mathrm{\Phi }}{g_0}$$

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	$\frac{m}{s^2}$
   $R$	local earth curvature radius	$m$
   $z$	altitude	$m$	altitude {:}
   $z_g$	geopotential height	$m$	geopotential_height {:}
   $\varphi$	latitude	$degN$	latitude {:}

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

   $$\begin{array}{rcl}g& =& 9.7803253359\frac{1+0.00193185265241{\sin }^2(\frac{\pi }{180}\varphi )}{\sqrt{1-0.00669437999013{\sin }^2(\frac{\pi }{180}\varphi )}}\\ R& =& \frac{1}{\sqrt{{\left(\frac{\cos (\frac{\pi }{180}\varphi )}{6356752.0}\right)}^2+{\left(\frac{\sin (\frac{\pi }{180}\varphi )}{6378137.0}\right)}^2}}\\ z_g& =& \frac{g}{g_0}\frac{Rz}{z+R}\end{array}$$

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	$\frac{kg{m}^2}{Kmol{s}^2}$
   $T(i)$	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 $p^B(1,1)$).

   $$\begin{array}{rcl}{z}_g(1)& =& z_{g,surf}+{10}^3\frac{T(1)}{M_{air}(1)}\frac{R}{g_0}\ln \left(\frac{p_{surf}}{p(i)}\right)\\ z_g(i)& =& z_g(i-1)+{10}^3\frac{T(i-1)+T(i)}{M_{air}(i-1)+M_{air}(i)}\frac{R}{g_0}\ln \left(\frac{p(i-1)}{p(i)}\right),1<i\le N\end{array}$$

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 {:}
   ${\mathrm{\Phi }}_{surf}$	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.

   $$z_{g,surf}=\frac{{\mathrm{\Phi }}_{surf}}{g_0}$$

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	$\frac{m}{s^2}$
   $R$	local earth curvature radius	$m$
   $z_{surf}$	surface altitude	$m$	surface_altitude {:}
   $z_{g,surf}$	surface geopotential height	$m$	surface_geopotential_height {:}
   $\varphi$	latitude	$degN$	latitude {:}

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

   $$\begin{array}{rcl}g& =& 9.7803253359\frac{1+0.00193185265241{\sin }^2(\frac{\pi }{180}\varphi )}{\sqrt{1-0.00669437999013{\sin }^2(\frac{\pi }{180}\varphi )}}\\ R& =& \frac{1}{\sqrt{{\left(\frac{\cos (\frac{\pi }{180}\varphi )}{6356752.0}\right)}^2+{\left(\frac{\sin (\frac{\pi }{180}\varphi )}{6378137.0}\right)}^2}}\\ z_{g,surf}& =& \frac{g}{g_0}\frac{R{z}_{surf}}{z_{surf}+R}\end{array}$$