Static atmospheric models describe how the
ideal gasproperties (namely: pressure, temperature, density, and molecular weight) of an atmosphere change, primarily as a function of altitude.
For example, the
US Standard Atmosphereis essentially a table of values for air temperature, pressure, and mass density, as a function of altitude above sea level.
Other static atmospheric models may have other outputs, or depend on inputs besides altitude.
The gas which comprises an atmosphere is usually assumed to be an
ideal gas, which is to say:
Where "ρ" is mass density, "M" is average molecular weight, "P" is pressure, "T" is temperature, and "R" is the ideal gas constant.
The gas is held in place by so called "
hydrostatic" forces. That is to say, for a particular layer of gas at some altitude: the downward (towards the planet) force of its weight, the downward force exerted by pressure in the layer above it, and the upward force exerted by pressure in the layer below, all sum to zero. Mathematically this is:
Finally, these variables describing the system do not change with time; i.e. it is a static system.
Depending on the model, some gas properties may be treated as constant with respect to altitude.
If the density of a gas is constant, then it isn't really behaving like a gas. Instead it is behaving like an
incompressible fluid, or liquid, and this situation looks more like an ocean. Assuming density is constant, then a graph of pressure vs altitude will have a constant slope, since the weight of the ocean above is directly proportional to its depth.
Isothermal-barotropic approximation and scale height
This atmospheric model assumes both molecular weight and temperature are constant over a wide range of altitude. Such a model may be called
isothermal(constant temperature). Inserting constant molecular weight and constant temperature into the equation for the ideal gas lawproduces the result that density and pressure, the two remaining variables, depend only on each other. For this reason, this model may also be called barotropic(density depends only on pressure).
For the isothermal-barotropic model, density and pressure turn out to be exponential functions of altitude. The increase in altitude necessary for "P" or "ρ" to drop by "1/e" is the called
where "R" is the ideal gas constant, "T" is temperature, "M" is average molecular weight, and "g"0 is the gravitational acceleration at the planet's surface. Using the values "T"=273 K and "M"=29 g/mol as characteristic of the Earth's atmosphere, "H" = "RT"/"Mg" = (8.315*273)/(29*9.8) = 7.99, or about 8 km, which coincidentally is approximate height of Mt. Everest.
For an isothermal atmosphere, or about 63% of the total mass of the atmosphere exists between the planet's surface and one scale height. (The total air mass below a certain altitude is calculated by integrating over the density function.)
For the ocean example there was a sharp transition in density at the top or "surface" of the ocean. However for atmospheres made of gas there is no equivalent sharp transition or edge. Gas atmospheres simply get less and less dense until they're so thin that they're space.
The U.S. Standard Atmosphere
The U.S. Standard Atmosphere model starts with many of the same assumptions as the isothermal-barotropic model, including ideal gas behavior, and constant molecular weight, but it differs by defining a more realistic temperature function, consisting of eight data points connected by straight lines; i.e. regions of constant temperature gradient. (See graph.) Of course the real atmosphere does not have a temperature distribution with this exact shape. The temperature function is an approximation. Values for pressure and density are then calculated based on this temperature function, and the constant temperature gradients help to make some of the math easier.
Gravitational acceleration, "g"(z), decreases with altitude since moving up means moving away from the planet's center.
This problem of decreasing "g" can be dealt with by defining a transformation from real geometric altitude "z" to an abstraction called "geopotential altitude" "h", defined: :
"h" has the property : where
Which basically says the amount of work done lifting a test mass "m" to height "z" through an atmosphere where gravity decreases with altitude, is the same as the amount of work done lifting that same mass to a height "h" through an atmosphere where "g" magically remains equal to "g0", its value at sea level. This geopotential altitude "h" is then used instead of geometric altitude "z" in the hydrostatic equations.
US Standard Atmosphereis one of the most common models
International Standard Atmosphereis a related international standard
NRLMSISE-00is a recent model from NRL often used in the atmospheric sciences
Jacchia Reference Atmosphereis an older model still commonly used in spacecraft dynamics
Standard conditions for temperature and pressure
Upper Atmospheric Models
* [http://www.pdas.com/hydro.pdf Public Domain Aeronautical Software - Derivation of hydrostatic equations used in the 1976 US Standard Atmosphere]
* [http://www.pdas.com/programs/atmos.f90 FORTRAN code to calculate the US Standard Atmosphere]
* [http://modelweb.gsfc.nasa.gov/atmos/about_atmos.html NASA GSFC Atmospheric Models overview]
* [http://modelweb.gsfc.nasa.gov/atmos/atmos_index.html Various models at NASA GSFC ModelWeb]
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