pressure derivations
- 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}. \[p = e^{\frac{ln(z^{B}(2)) + ln(z^{B}(1))}{2}}\]
- pressure from altitude - symbol - description - unit - variable name - \(a\) - WGS84 semi-major axis - \(m\) - \(b\) - WGS84 semi-minor axis - \(m\) - \(f\) - WGS84 flattening - \(m\) - \(g\) - gravity - \(\frac{m}{s^2}\) - \(g_{0}\) - mean earth gravity - \(\frac{m}{s^2}\) - \(g_{surf}\) - gravity at surface - \(\frac{m}{s^2}\) - \(GM\) - 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_{surf} & = & 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_{surf} \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_{surf} \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(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(i)}{R}\left(z(i)-z(i-1)\right)}, 1 < i \leq N \end{eqnarray}
- 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}
- 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. \[p = nkT\]
- 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. \[p_{surf} = n_{surf}kT_{surf}\]
- 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 \\ & \forall_{j, i < j <= N \wedge z(j)-z(i) <= 2000} \frac{T(i)-T(j)}{z(j)-z(i)} <= 0.002 & \end{eqnarray}- If no such \(i\) can be found then \(p_{TP}\) is set to NaN.