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Common formula
==============
gravity
-------
#. Newton's gravitational law
================ ============================ ======================
symbol description unit
================ ============================ ======================
:math:`g` gravity :math:`\frac{m}{s^2}`
:math:`g_{surf}` gravity at earth surface :math:`\frac{m}{s^2}`
:math:`h` height above surface :math:`m`
:math:`R_{surf}` earth radius at surface :math:`m`
================ ============================ ======================
.. math::
g = g_{surf}\left(\frac{R_{surf}}{R_{surf} + h}\right)^2
#. normal gravity at ellipsoid surface
This is the WGS84 ellipsoidal gravity formula as taken from NIMA TR8350.2
================ ===================== =====================
symbol name unit
================ ===================== =====================
:math:`a` WGS84 semi-major axis :math:`m`
:math:`b` WGS84 semi-minor axis :math:`m`
:math:`e` eccentricity :math:`m`
:math:`g_{e}` gravity at equator :math:`\frac{m}{s^2}`
:math:`g_{p}` gravity at poles :math:`\frac{m}{s^2}`
:math:`g_{surf}` gravity at surface :math:`\frac{m}{s^2}`
:math:`\phi` latitude :math:`degN`
================ ===================== =====================
.. math::
:nowrap:
\begin{eqnarray}
e^2 & = & \frac{a^2-b^2}{a^2} \\
k & = & \frac{bg_{p} - ag_{e}}{ag_{e}} \\
g_{surf} & = & g_{e}\frac{1 + k {\sin}^2(\frac{\pi}{180}\phi)}{\sqrt{1 - e^2{\sin}^2(\frac{\pi}{180}\phi)}} \\
g_{surf} & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)}
{\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}}
\end{eqnarray}
#. normal gravity above ellipsoid surface
This is the WGS84 ellipsoidal gravity formula as taken from NIMA TR8350.2
================ ==================================== =======================
symbol name unit
================ ==================================== =======================
:math:`a` WGS84 semi-major axis :math:`m`
:math:`b` WGS84 semi-minor axis :math:`m`
:math:`f` WGS84 flattening :math:`m`
:math:`g` gravity :math:`\frac{m}{s^2}`
:math:`g_{surf}` gravity at the ellipsoid surface :math:`\frac{m}{s^2}`
:math:`GM` WGS84 earth's gravitational constant :math:`\frac{m^3}{s^2}`
:math:`z` altitude :math:`m`
:math:`\phi` latitude :math:`degN`
:math:`\omega` WGS84 earth angular velocity :math:`rad/s`
================ ==================================== =======================
The formula used is the one based on the truncated Taylor series expansion:
.. math::
:nowrap:
\begin{eqnarray}
m & = & \frac{\omega^2a^2b}{GM} \\
g & = & g_{surf} \left[ 1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)z + \frac{3}{a^2}z^2 \right] \\
\end{eqnarray}
geopotential height
-------------------
================ ============================ ======================
symbol description unit
================ ============================ ======================
:math:`g` gravity :math:`\frac{m}{s^2}`
:math:`g_{0}` mean earth gravity :math:`\frac{m}{s^2}`
:math:`g_{surf}` gravity at earth surface :math:`\frac{m}{s^2}`
:math:`p` pressure :math:`Pa`
:math:`R_{surf}` earth radius at surface :math:`m`
:math:`h` height above surface :math:`m`
:math:`h_{g}` geopotential height :math:`m`
:math:`\phi` latitude :math:`degN`
:math:`\rho` mass density :math:`\frac{kg}{m^3}`
================ ============================ ======================
The geopotential height allows the gravity in the hydrostatic equation
.. math::
dp = - \rho g dh
to be replaced by a constant gravity
.. math::
dp = - \rho g_{0} dh_{g}
providing
.. math::
dh_{g} = \frac{g}{g_{0}}dh
With Newton's gravitational law this becomes
.. math::
dh_{g} = \frac{g_{surf}}{g_{0}}\left(\frac{R_{surf}}{R_{surf} + h}\right)^2dh
And integrating this, considering that :math:`h=0` and :math:`h_{g}=0` at the surface, results in
.. math::
h_{g} = \frac{g_{surf}}{g_{0}}\frac{R_{surf}h}{R_{surf} + h}
.. math::
h = \frac{g_{0}R_{surf}h_{g}}{g_{surf}R_{surf}-g_{0}h_{g}}
gas constant
------------
=========== ====================== ================================
symbol name unit
=========== ====================== ================================
:math:`k` Boltzmann constant :math:`\frac{kg m^2}{K s^2}`
:math:`N_A` Avogadro constant :math:`\frac{1}{mol}`
:math:`R` universal gas constant :math:`\frac{kg m^2}{K mol s^2}`
=========== ====================== ================================
Relation between Boltzmann constant, universal gas constant, and Avogadro constant:
.. math::
k = \frac{R}{N_A}
ideal gas law
-------------
========= ====================== ================================
symbol name unit
========= ====================== ================================
:math:`k` Boltzmann constant :math:`\frac{kg m^2}{K s^2}`
:math:`N` amount of substance :math:`molec`
:math:`p` pressure :math:`Pa`
:math:`R` universal gas constant :math:`\frac{kg m^2}{K mol s^2}`
:math:`T` temperature :math:`K`
:math:`V` volume :math:`m^3`
========= ====================== ================================
.. math::
pV = \frac{NRT}{N_{A}} = NkT
barometric formula
------------------
=============== ======================= ================================
symbol name unit
=============== ======================= ================================
:math:`g` gravity :math:`\frac{m}{s^2}`
:math:`g_{0}` mean earth gravity :math:`\frac{m}{s^2}`
:math:`k` Boltzmann constant :math:`\frac{kg m^2}{K s^2}`
:math:`M_{air}` molar mass of total air :math:`\frac{g}{mol}`
:math:`N` amount of substance :math:`molec`
:math:`N_A` Avogadro constant :math:`\frac{1}{mol}`
:math:`p` pressure :math:`Pa`
:math:`R` universal gas constant :math:`\frac{kg m^2}{K mol s^2}`
:math:`T` temperature :math:`K`
:math:`V` volume :math:`m^3`
:math:`z` altitude :math:`m`
:math:`z_{g}` geopotential height :math:`m`
:math:`\phi` latitude :math:`degN`
:math:`\rho` mass density :math:`\frac{kg}{m^3}`
=============== ======================= ================================
From the ideal gas law we have:
.. math::
p = \frac{NkT}{V} = \frac{10^{-3}NM_{air}}{VN_{a}}\frac{kTN_{a}}{10^{-3}M_{air}} = \rho\frac{RT}{10^{-3}M_{air}}
And from the hydrostatic assumption we get:
.. math::
dp = - \rho g dz
Dividing :math:`dp` by `p` we get:
.. math::
\frac{dp}{p} = -\frac{10^{-3}M_{air}\rho g dz}{\rho RT} = -\frac{10^{-3}M_{air}gdz}{RT}
Integrating this expression from one pressure level to the next we get:
.. math::
p(i+1) = p(i)e^{-\int^{z(i+1)}_{z(i)}\frac{10^{-3}M_{air}g}{RT}dz}
We can approximate this further by using an average value of the height dependent quantities
:math:`M_{air}`, :math:`g` and :math:`T` for the integration over the range :math:`[z(i),z(i+1)]`.
This gives:
.. math::
:nowrap:
\begin{eqnarray}
g & = & g(\phi,\frac{z(i)+z(i+1)}{2}) \\
p(i+1) & = & p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{2}\frac{2}{T(i)+T(i+1)}\frac{g}{R}\left(z(i+1)-z(i)\right)} \\
& = & p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{T(i)+T(i+1)}\frac{g}{R}\left(z(i+1)-z(i)\right)}
\end{eqnarray}
When using geopotential height the formula is the same except that :math:`g=g_{0}` at all levels:
.. math::
p(i+1) = p(i)e^{-10^{-3}\frac{M_{air}(i)+M_{air}(i+1)}{T(i)+T(i+1)}\frac{g_{0}}{R}\left(z_{g}(i+1)-z_{g}(i)\right)}
mass density
------------
=============== ======================= ======================
symbol name unit
=============== ======================= ======================
:math:`N` amount of substance :math:`molec`
:math:`N_A` Avogadro constant :math:`\frac{1}{mol}`
:math:`M_{air}` molar mass of total air :math:`\frac{g}{mol}`
:math:`V` volume :math:`m^3`
:math:`\rho` mass density :math:`\frac{kg}{m^3}`
=============== ======================= ======================
.. math::
\rho = \frac{10^{-3}NM_{air}}{VN_{a}}
number density
--------------
========= =================== =========================
symbol name unit
========= =================== =========================
:math:`n` number density :math:`\frac{molec}{m^3}`
:math:`N` amount of substance :math:`molec`
:math:`V` volume :math:`m^3`
========= =================== =========================
.. math::
n = \frac{N}{V}
dry air vs. total air
---------------------
======================= =========================== =========================
symbol name unit
======================= =========================== =========================
:math:`n` number density of total air :math:`\frac{molec}{m^3}`
:math:`n_{dry\_air}` number density of dry air :math:`\frac{molec}{m^3}`
:math:`n_{H_{2}O}` number density of H2O :math:`\frac{molec}{m^3}`
:math:`M_{air}` molar mass of total air :math:`\frac{g}{mol}`
:math:`M_{dry\_air}` molar mass of dry air :math:`\frac{g}{mol}`
:math:`M_{H_{2}O}` molar mass of H2O :math:`\frac{g}{mol}`
:math:`\rho` mass density of total air :math:`\frac{kg}{m^3}`
:math:`\rho_{dry\_air}` mass density of dry air :math:`\frac{kg}{m^3}`
:math:`\rho_{H_{2}O}` mass density of H2O :math:`\frac{kg}{m^3}`
======================= =========================== =========================
.. math::
:nowrap:
\begin{eqnarray}
n & = & n_{dry\_air} + n_{H_{2}O} \\
M_{air}n & = & M_{dry\_air}n_{dry\_air} + M_{H_{2}O}n_{H_{2}O} \\
\rho & = & \rho_{dry\_air} + \rho_{H_{2}O} \\
\end{eqnarray}
virtual temperature
-------------------
==================== ======================== ================================
symbol name unit
==================== ======================== ================================
:math:`k` Boltzmann constant :math:`\frac{kg m^2}{K s^2}`
:math:`M_{air}` molar mass of total air :math:`\frac{g}{mol}`
:math:`M_{dry\_air}` molar mass of dry air :math:`\frac{g}{mol}`
:math:`M_{H_{2}O}` molar mass of H2O :math:`\frac{g}{mol}`
:math:`N` amount of substance :math:`molec`
:math:`N_A` Avogadro constant :math:`\frac{1}{mol}`
:math:`p` pressure :math:`Pa`
:math:`p_{dry\_air}` dry air partial pressure :math:`Pa`
:math:`p_{H_{2}O}` H2O partial pressure :math:`Pa`
:math:`R` universal gas constant :math:`\frac{kg m^2}{K mol s^2}`
:math:`T` temperature :math:`K`
:math:`T_{v}` virtual temperature :math:`K`
:math:`V` volume :math:`m^3`
==================== ======================== ================================
From the ideal gas law we have:
.. math::
p = \frac{NkT}{V} = \frac{10^{-3}NM_{air}}{VN_{a}}\frac{kTN_{a}}{10^{-3}M_{air}} = \rho \frac{RT}{10^{-3}M_{air}}
The virtual temperature allows us to use the dry air molar mass in this equation:
.. math::
p = \rho\frac{RT_{v}}{10^{-3}M_{dry\_air}}
This gives:
.. math::
T_{v} = \frac{M_{dry\_air}}{M_{air}}T
volume mixing ratio
-------------------
====================== =============================== =========================
symbol name unit
====================== =============================== =========================
:math:`n` number density of total air :math:`\frac{molec}{m^3}`
:math:`n_{dry\_air}` number density of dry air :math:`\frac{molec}{m^3}`
:math:`n_{H_{2}O}` number density of H2O :math:`\frac{molec}{m^3}`
:math:`n_{x}` number density of quantity x :math:`\frac{molec}{m^3}`
:math:`\nu_{x}` volume mixing ratio of quantity :math:`ppv`
x with regard to total air
:math:`\bar{\nu}_{x}` volume mixing ratio of quantity :math:`ppv`
x with regard to dry air
====================== =============================== =========================
.. math::
:nowrap:
\begin{eqnarray}
\nu_{x} & = & \frac{n_{x}}{n} \\
\bar{\nu}_{x} & = & \frac{n_{x}}{n_{dry\_air}} \\
\nu_{dry\_air} & = & \frac{n_{dry\_air}}{n} =
\frac{n - n_{H_{2}O}}{n} = 1 - \nu_{H_{2}O} \\
\nu_{air} & = & \frac{n}{n} = 1 \\
\bar{\nu}_{dry\_air} & = & \frac{n_{dry\_air}}{n_{dry\_air}} = 1 \\
\bar{\nu}_{H_{2}O} & = & \frac{n_{H_{2}O}}{n_{dry\_air}} =
\frac{\nu_{H_{2}O}}{\nu_{dry\_air}} = \frac{\nu_{H_{2}O}}{1 - \nu_{H_{2}O}} \\
\nu_{H_{2}O} & = & \frac{\bar{\nu}_{H_{2}O}}{1 + \bar{\nu}_{H_{2}O}}
\end{eqnarray}
mass mixing ratio
-----------------
===================== =============================== =========================
symbol name unit
===================== =============================== =========================
:math:`M_{air}` molar mass of total air :math:`\frac{g}{mol}`
:math:`M_{dry\_air}` molar mass of dry air :math:`\frac{g}{mol}`
:math:`M_{x}` molar mass of quantity x :math:`\frac{g}{mol}`
:math:`n` number density of total air :math:`\frac{molec}{m^3}`
:math:`n_{dry\_air}` number density of dry air :math:`\frac{molec}{m^3}`
:math:`n_{H_{2}O}` number density of H2O :math:`\frac{molec}{m^3}`
:math:`n_{x}` number density of quantity x :math:`\frac{molec}{m^3}`
:math:`q_{x}` mass mixing ratio of quantity x :math:`\frac{kg}{kg}`
with regard to total air
:math:`\bar{q}_{x}` mass mixing ratio of quantity x :math:`\frac{kg}{kg}`
with regard to dry air
:math:`\nu_{x}` volume mixing ratio of quantity :math:`ppv`
x with regard to total air
:math:`\bar{\nu}_{x}` volume mixing ratio of quantity :math:`ppv`
x with regard to dry air
===================== =============================== =========================
.. math::
:nowrap:
\begin{eqnarray}
q_{x} & = & \frac{n_{x}M_{x}}{nM_{air}} = \nu_{x}\frac{M_{x}}{M_{air}} \\
\bar{q}_{x} & = & \frac{n_{x}M_{x}}{n_{dry\_air}M_{dry\_air}} = \bar{\nu}_{x}\frac{M_{x}}{M_{dry\_air}} \\
q_{dry\_air} & = & \frac{n_{dry\_air}M_{dry\_air}}{nM_{air}} =
\frac{nM_{air} - n_{H_{2}O}M_{H_{2}O}}{nM_{air}} = 1 - q_{H_{2}O} \\
q_{air} & = & \frac{nM_{air}}{nM_{air}} = 1 \\
\bar{q}_{dry\_air} & = & \frac{n_{dry\_air}M_{dry\_air}}{n_{dry\_air}M_{dry\_air}} = 1 \\
\bar{q}_{H_{2}O} & = & \frac{n_{H_{2}O}M_{H_{2}O}}{n_{dry\_air}M_{dry\_air}} =
\frac{q_{H_{2}O}}{q_{dry\_air}} = \frac{q_{H_{2}O}}{1 - q_{H_{2}O}} \\
q_{H_{2}O} & = & \frac{\bar{q}_{H_{2}O}}{1 + \bar{q}_{H_{2}O}}
\end{eqnarray}
molar mass of total air
-----------------------
#. molar mass of total air from H2O volume mixing ratio
==================== =========================== =========================
symbol name unit
==================== =========================== =========================
:math:`M_{air}` molar mass of total air :math:`\frac{g}{mol}`
:math:`M_{dry\_air}` molar mass of dry air :math:`\frac{g}{mol}`
:math:`M_{H_{2}O}` molar mass of H2O :math:`\frac{g}{mol}`
:math:`n` number density of total air :math:`\frac{molec}{m^3}`
:math:`n_{dry\_air}` number density of dry air :math:`\frac{molec}{m^3}`
:math:`n_{H_{2}O}` number density of H2O :math:`\frac{molec}{m^3}`
:math:`\nu_{H_{2}O}` volume mixing ratio of H2O :math:`ppv`
==================== =========================== =========================
.. math::
:nowrap:
\begin{eqnarray}
M_{air} & = & \frac{M_{dry\_air}n_{dry\_air} + M_{H_{2}O}n_{H_{2}O}}{n} \\
& = & M_{dry\_air}\left(1 - \nu_{H_{2}O}\right) + M_{H_{2}O}\nu_{H_{2}O}
\end{eqnarray}
#. molar mass of total air from H2O mass mixing ratio
==================== =========================== =========================
symbol name unit
==================== =========================== =========================
:math:`M_{air}` molar mass of total air :math:`\frac{g}{mol}`
:math:`M_{dry\_air}` molar mass of dry air :math:`\frac{g}{mol}`
:math:`M_{H_{2}O}` molar mass of H2O :math:`\frac{g}{mol}`
:math:`n` number density of total air :math:`\frac{molec}{m^3}`
:math:`n_{dry\_air}` number density of dry air :math:`\frac{molec}{m^3}`
:math:`n_{H_{2}O}` number density of H2O :math:`\frac{molec}{m^3}`
:math:`q_{H_{2}O}` mass mixing ratio of H2O :math:`\frac{kg}{kg}`
:math:`\nu_{H_{2}O}` volume mixing ratio of H2O :math:`\frac{kg}{kg}`
==================== =========================== =========================
.. math::
:nowrap:
\begin{eqnarray}
M_{air} & = & M_{dry\_air}\left(1 - \nu_{H_{2}O}\right) + M_{H_{2}O}\nu_{H_{2}O} \\
& = & M_{dry\_air}\left(1 - \frac{M_{air}}{M_{H_{2}O}}q_{H_{2}O}\right) + M_{air}q_{H_{2}O} \\
& = & \frac{M_{dry\_air}}{1 + \frac{M_{dry\_air}}{M_{H_{2}O}}q_{H_{2}O} - q_{H_{2}O}} \\
& = & \frac{M_{H_{2}O}M_{dry\_air}}{M_{H_{2}O} + M_{dry\_air}q_{H_{2}O} - M_{H_{2}O}q_{H_{2}O}} \\
& = & \frac{M_{H_{2}O}M_{dry\_air}}{\left(1-q_{H_{2}O}\right)M_{H_{2}O} + q_{H_{2}O}M_{dry\_air}} \\
\end{eqnarray}
partial pressure
----------------
===================== =============================== ===========
symbol name unit
===================== =============================== ===========
:math:`p` total pressure :math:`Pa`
:math:`p_{x}` partial pressure of quantity :math:`Pa`
:math:`\nu_{x}` volume mixing ratio of quantity :math:`ppv`
x with regard to total air
:math:`\bar{\nu}_{x}` volume mixing ratio of quantity :math:`ppv`
x with regard to dry air
===================== =============================== ===========
.. math::
:nowrap:
\begin{eqnarray}
p_{x} & = & \nu_{x}p \\
p_{x} & = & \bar{\nu}_{x}p_{dry\_air} \\
p_{x} & = & N_{x}kT
\end{eqnarray}
saturated water vapor pressure
------------------------------
============= =============================== ===========
symbol name unit
============= =============================== ===========
:math:`e_{w}` saturated water vapor pressure :math:`Pa`
:math:`T` temperature :math:`K`
============= =============================== ===========
This is the August-Roche-Magnus formula for the saturated water vapour pressure
.. math::
e_{w} = 610.94e^{\frac{17.625(T-273.15)}{(T-273.15)+243.04}}
|
