Editing Lapse rate (section)

== Mathematics of the adiabatic lapse rate==
[[File:Adiabatic_lapse_rate.svg|thumb|Simplified graph of atmospheric lapse rate near sea level]]
The following calculations derive the temperature as a function of altitude for a packet of air which is ascending or descending without exchanging heat with its environment.  

===Dry adiabatic lapse rate===
Thermodynamics defines an adiabatic process as:
:<math>P\, \mathrm{d}V = -\frac{V\, \mathrm{d}P}{\gamma}</math>

the [[first law of thermodynamics]] can be written as
:<math>m c_\text{v}\, \mathrm{d}T - \frac{V \,\mathrm{d}P}{\gamma} = 0</math>

Also, since the density <math>\rho = m/V</math> and <math>\gamma = c_\text{p}/c_\text{v}</math>, we can show that:
:<math>\rho c_\text{p} \,\mathrm{d}T - \mathrm{d}P = 0</math>

where <math>c_\text{p}</math> is the [[specific heat]] at constant pressure.

Assuming an atmosphere in [[hydrostatic equilibrium]]:<ref name="LL">Landau and Lifshitz, ''Fluid Mechanics'', Pergamon, 1979</ref>
:<math>\mathrm{d}P = -\rho g\, \mathrm{d}z</math>

where ''g'' is the [[standard gravity]]. Combining these two equations to eliminate the pressure, one arrives at the result for the ''dry adiabatic lapse rate'' (DALR),<ref>{{cite book|last1=Kittel|last2=Kroemer|title=Thermal Physics|publisher=W. H. Freeman|year=1980|chapter-url=https://books.google.com/books?id=c0R79nyOoNMC&pg=PA179|chapter=6|isbn=978-0-7167-1088-2|page=179}} problem 11</ref>
:<math>\Gamma_\text{d} = -\frac{\mathrm{d}T}{\mathrm{d}z} = \frac{g}{c_\text{p}} = 9.8\ ^{\circ}\text{C}/\text{km}</math>

The DALR (<math>\Gamma_\text{d}</math>) is the temperature gradient experienced in an ascending or descending packet of air that is not saturated with water vapor, i.e., with less than 100% relative humidity. 

=== Moist adiabatic lapse rate  ===

The presence of water within the atmosphere (usually the troposphere) complicates the process of convection. Water vapor contains latent [[heat of vaporization]]. As a parcel of air rises and cools, it eventually becomes [[Dew point|saturated]]; that is, the vapor pressure of water in equilibrium with liquid water has decreased (as temperature has decreased) to the point where it is equal to the actual vapor pressure of water. With further decrease in temperature the water vapor in excess of the equilibrium amount condenses, forming [[cloud]], and releasing heat (latent heat of condensation). Before saturation, the rising air follows the dry adiabatic lapse rate. After saturation, the rising air follows the moist (or ''wet'') adiabatic lapse rate.<ref>{{cite web|url=http://meteorologytraining.tpub.com/14312/css/14312_47.htm |title=Dry Adiabatic Lapse Rate |publisher=tpub.com |access-date=2016-05-02 |url-status=dead |archive-url=https://web.archive.org/web/20160603041448/http://meteorologytraining.tpub.com/14312/css/14312_47.htm |archive-date=2016-06-03 }}</ref> The release of latent heat is an important source of energy in the development of thunderstorms.

While the dry adiabatic lapse rate is a constant {{nowrap|9.8&nbsp;°C/km}} ({{nowrap|5.4&nbsp;°F}} per 1,000&nbsp;ft, {{nowrap|3&nbsp;°C/1,000&nbsp;ft}}), the moist adiabatic lapse rate varies strongly with temperature. A typical value is around {{nowrap|5&nbsp;°C/km}}, ({{nowrap|9&nbsp;°F/km}}, {{nowrap|2.7&nbsp;°F/1,000&nbsp;ft}}, {{nowrap|1.5&nbsp;°C/1,000&nbsp;ft}}).<ref name="MLM">{{cite journal|last1=Minder|first1=JR|first2=PW|last2=Mote|first3=JD|last3=Lundquist|year=2010|title=Surface temperature lapse rates over complex terrain: Lessons from the Cascade Mountains|journal=J. Geophys. Res.|volume=115|issue=D14|page=D14122|doi=10.1029/2009JD013493|bibcode = 2010JGRD..11514122M |doi-access=free}}</ref> The formula for the ''saturated adiabatic lapse rate'' (SALR) or ''moist adiabatic lapse rate'' (MALR) is given by:<ref>{{cite web|url=http://glossary.ametsoc.org/wiki/Saturation-adiabatic_lapse_rate|publisher=American Meteorological Society|title=Saturation adiabatic lapse rate|work=Glossary}}</ref>

:<math>\Gamma_\text{w}
  = g\, \frac{\left(1 + \dfrac{H_\text{v}\, r}{R_\text{sd}\, T}\right)}{\left(c_\text{pd} + \dfrac{H_\text{v}^2\, r}{R_\text{sw}\, T^2}\right)}
</math>

where:
:{| border="0" cellpadding="2"
|-
| style="text-align:right;" | <math>\Gamma_\text{w}</math>,
| wet adiabatic lapse rate, K/m
|-
| style="text-align:right;" | <math>g</math>,
| Earth's [[Standard gravity|gravitational acceleration]] = 9.8076&nbsp;m/s<sup>2</sup>
|-
| style="text-align:right;" | <math>H_v</math>,
| [[heat of vaporization]] of water = {{val|2501000|u=J/kg}} 
|-
| style="text-align:right;" | <math>R_\text{sd}</math>,
| [[specific gas constant]] of dry air = 287&nbsp;J/kg·K 
|-
| style="text-align:right;" | <math>R_\text{sw}</math>,
| specific gas constant of water vapour = 461.5&nbsp;J/kg·K
|-
| style="text-align:right;" | <math>\epsilon = \frac{R_\text{sd}}{R_\text{sw}}</math>,
| the dimensionless ratio of the specific gas constant of dry air to the specific gas constant for water vapour = 0.622
|-
| style="text-align:right;" | <math>e</math>,
| the water [[vapour pressure]] of the saturated air
|-
| style="text-align:right;" | <math>r = \frac{\epsilon e}{p - e}</math>,
| the [[mixing ratio]] of the mass of water vapour to the mass of dry air<ref>{{cite web|url=http://glossary.ametsoc.org/wiki/Mixing_ratio|publisher=American Meteorological Society|title=Mixing ratio|work=Glossary}}</ref>
|-
| style="text-align:right;" | <math>p</math>,
| the pressure of the saturated air
|-
|-
| style="text-align:right;" | <math>T</math>,
| temperature of the saturated air, K
|-
| style="text-align:right;" | <math>c_\text{pd}</math>,
| the [[specific heat]] of dry air at constant pressure, = 1003.5{{nbsp}}J/kg·K 
|}

The SALR or MALR (<math>\Gamma_\text{w}</math>) is the temperature gradient experienced in an ascending or descending packet of air that is saturated with water vapor, i.e., with 100% relative humidity.