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| In [[atmospheric thermodynamics]], the '''virtual temperature''' <math>T_v</math> of a moist [[air parcel]] is the [[temperature]] at which a theoretical dry [[air]] parcel would have a total [[pressure]] and [[density]] equal to the moist parcel of air.<ref>{{cite book
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| | last = Bailey
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| | first = Desmond T.
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| | others = John Irwin
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| | title = Meteorological Monitoring Guidance for Regulatory Modeling Applications
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| | origyear = 1987
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| | origmonth = 6
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| | url = http://www.epa.gov/scram001/guidance/met/mmgrma.pdf
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| | year = 2000
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| | month = 2
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| | publisher = [[United States Environmental Protection Agency]]
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| | location = Research Triangle Park, NC
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| | id = EPA-454/R-99-005
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| | pages = 9–14
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| | chapter = Upper-air Monitoring
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| }}</ref>
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| ==Introduction==
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| ===Description===
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| In atmospheric [[thermodynamic process]]es, it is often useful to assume air parcels behave approximately [[adiabatic process|adiabatic]], and thus approximately [[Ideal gas law|ideally]]. The [[Gas constant#Specific gas constant|gas constant]] for the standardized mass of one kilogram of a particular gas is dynamic, and described mathematically as:
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| <math>{R_x}=1000\frac{R^{*}}{M_x} \, ,</math>
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| where <math>R^{*}</math> is the universal gas constant and <math>M_x</math> is the apparent [[molecular weight]] of gas <math>x</math>. The apparent molecular weight of a theoretical moist parcel in [[Earth's atmosphere]] can be defined in components of dry and moist air as:
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| <math>{M_{air}}=\frac{e}{p}M_v+\frac{p_d}{p}M_d \, ,</math>
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| with <math>e</math> [[water vapor pressure]], <math>p_d</math> dry [[air pressure]], and <math>M_v</math> and <math>M_d</math> representing the molecular weight of water and dry air respectively. The total pressure <math>p</math> is described by [[Dalton's Law|Dalton's Law of Partial Pressures]]:
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| <math>{p}={p_d}+{e} \, .</math>
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| ===Purpose===
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| Rather than carry out these calculations, it is convenient to scale another quantity within the ideal gas law to equate the pressure and density of a dry parcel to a moist parcel. The only variable quantity of the ideal gas law independent of density and pressure is temperature. This scaled quantity is known as virtual temperature, and it allows for the use of the dry-air [[equation of state]] for moist air.<ref>{{cite web
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| |url=http://amsglossary.allenpress.com/glossary/search?id=virtual-temperature1
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| |title=AMS Glossary
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| |accessdate=2007-05-07
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| |publisher=American Meteorological Society
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| }}</ref> Temperature has an inverse proportionality to density. Thus, analytically, a higher vapor pressure would yield a lower density, which should yield a higher virtual temperature in turn.
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| ==Derivation==
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| Consider an air parcel containing masses <math>m_d</math> and <math>m_v</math> of water vapor in a given volume <math>V</math>. The density is given by:
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| <math>{\rho}=\frac{m_d+m_v}{V}=\rho_d+\rho_v \, ,</math>
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| where <math>\rho_d</math> and <math>\rho_v</math> are the densities of dry air and water vapor would respectively have when occupying the volume of the air parcel. Rearranging the standard ideal gas equation with these variables gives:
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| <math>{e}=\rho_vR_vT \,</math> and <math>{p_d}=\rho_dR_dT \, .</math>
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| Solving for the densities in each equation and combining with the law of partial pressures yields:
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| <math>{\rho}=\frac{p-e}{R_dT}+\frac{e}{R_vT}\, .</math>
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| Then, solving for <math>p</math> and using <math>\textstyle\epsilon=\frac{R_d}{R_v}=\frac{M_v}{M_d}</math> is approximately 0.622 in Earth's atmosphere:
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| <math>{p}={\rho}R_dT_v \, ,</math>
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| where the virtual temperature <math>T_v</math> is:
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| <math>{T_v}=\frac{T}{1-\frac{e}{p}(1-{\epsilon})}\, .</math>
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| We now have a non-linear [[Scalar (mathematics)|scalar]] for temperature dependent purely on the [[unitless]] value <math>\scriptstyle\frac{e}{p}\, ,</math> allowing for varying amounts of water vapor in an air parcel. This virtual temperature <math>T_v</math> in units of [[Kelvin]] can be used seamlessly in any thermodynamic equation necessitating it.
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| ==Variations==
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| Often the more easily accessible atmospheric parameter is the [[mixing ratio#Mixing ratio or humidity ratio|mixing ratio]] <math>w</math>. Through expansion upon the definition of vapor pressure in the law of partial pressures as presented above and the definition of mixing ratio:
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| <math>\frac{e}{p}=\frac{w}{w+{\epsilon}}\, ,</math>
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| which allows:
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| <math>{T_v}=T\frac{w+\epsilon}{\epsilon(1+w)}\, .</math>
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| Algebraic expansion of that equation, ignoring higher orders of <math>w</math> due to its typical order in Earth's atmosphere of <math>10^{-3}</math>, and substituting <math>\epsilon</math> with its constant value yields the linear approximation:
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| <math>{T_v} \approx T(1+0.61w)\, .</math>
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| An approximate conversion using <math>T</math> in degrees [[Celsius]] and mixing ratio <math>w</math> in g/kg is:
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| <math>{T_v} \approx T+\frac{w}{6}\, .</math><ref>{{cite book
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| | last = U.S. Air Force
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| | first =
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| | others =
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| | title = The Use of the Skew-T Log p Diagram in Analysis and Forecasting
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| | origyear = 1990
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| | origmonth =
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| | year = 1990
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| | publisher = [[United States Air Force]]
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| | id = AWS-TR79/006
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| | pages = 4–9
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| }}</ref>
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| ==Uses==
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| Virtual temperature is used in adjusting [[Convective available potential energy|CAPE]] soundings for assessing available convective potential energy from [[Skew-T log-P diagram]]s. The errors associated with ignoring virtual temperature correction for smaller CAPE values can be quite significant.<ref>{{cite web
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| |url=http://journals.ametsoc.org/doi/abs/10.1175/1520-0434%281994%29009%3C0625%3ATEONTV%3E2.0.CO%3B2
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| |title=The Effect of Neglecting the Virtual Temperature Correction on CAPE calculations, Weather and Forecasting 1994; 9: 625-629
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| |accessdate=2010-06-02
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| |publisher=American Meteorological Society
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| }}</ref> Thus, in the early stages of convective storm formation, a virtual temperature correction is significant in identifying the [[maximum potential intensity|potential intensity]] in [[tropical cyclogenesis]].<ref>{{cite web
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| |url=http://www3.interscience.wiley.com/cgi-bin/fulltext/118494791/HTMLSTART
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| |title=Tropical cyclone genesis potential index in climate models
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| |accessdate=2009-12-10
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| |publisher=International Research Institute for Climate and Society
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| }}</ref>
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| ==Further reading==
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| *{{cite book
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| | last1 = Wallace
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| | first1 = John M.
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| | first2 = Peter V.
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| | last2 = Hobbs
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| | title = Atmospheric Science
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| | year = 2006
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| | isbn = 0-12-732951-X}}
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Virtual Temperature}}
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| [[Category:Atmospheric thermodynamics]]
| |
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