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In [[atmospheric science]], the '''
▲In [[atmospheric science]], the '''Weak Temperature Gradient approximation''' (WTG) is a theoretical framework used to simplify the equations governing tropical atmospheric dynamics and circulation. The WTG approximation assumes that free [[Troposphere|tropospheric]] temperature in the [[tropics]] has negligible horizontal (and temporal) gradients compared to its vertical gradient.<ref name=":0">{{Cite journal |last=Raymond |first=David J. |last2=Zeng |first2=Xiping |date=2005-04-01 |title=Modelling tropical atmospheric convection in the context of the weak temperature gradient approximation |url=http://doi.wiley.com/10.1256/qj.03.97 |journal=Quarterly Journal of the Royal Meteorological Society |language=en |volume=131 |issue=608 |pages=1301–1320 |doi=10.1256/qj.03.97}}</ref><ref name=":1">{{Cite journal |last=Sobel |first=Adam H. |last2=Bretherton |first2=Christopher S. |date=2000-12-15 |title=Modeling Tropical Precipitation in a Single Column |url=https://journals.ametsoc.org/view/journals/clim/13/24/1520-0442_2000_013_4378_mtpias_2.0.co_2.xml |journal=Journal of Climate |language=EN |volume=13 |issue=24 |pages=4378–4392 |doi=10.1175/1520-0442(2000)013<4378:MTPIAS>2.0.CO;2 |issn=0894-8755}}</ref>
The assumption of horizontal homogeneity of temperature follows from observations of free tropospheric temperature in the tropical regions as well as early work on the simplified equations governing tropical circulation. It is understood to occur as a result of the weak [[Coriolis force]] in the tropics.<ref name=":2">{{Cite book |url=https://www.cambridge.org/core/books/clouds-and-climate/7B47159F7B050B71625111E40795D182 |title=Clouds and Climate: Climate Science's Greatest Challenge |date=2020 |publisher=Cambridge University Press |isbn=978-1-107-06107-1 |editor-last=Siebesma |editor-first=A. Pier |___location=Cambridge |editor-last2=Bony |editor-first2=Sandrine |editor-last3=Jakob |editor-first3=Christian |editor-last4=Stevens |editor-first4=Bjorn}}</ref><ref name=":3">{{Cite journal |last=Charney |first=Jule G. |date=1963-11-01 |title=A Note on Large-Scale Motions in the Tropics |url=https://journals.ametsoc.org/view/journals/atsc/20/6/1520-0469_1963_020_0607_anolsm_2_0_co_2.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=20 |issue=6 |pages=607–609 |doi=10.1175/1520-0469(1963)020<0607:ANOLSM>2.0.CO;2 |issn=0022-4928}}</ref>
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In a multitude of theoretical, modelling and observational studies, the WTG has been applied to study [[Synoptic scale meteorology|synoptic]]- and [[Mesoscale meteorology|mesoscale]] phenomena in the tropics.
== Physical
Free tropospheric temperature refers to the [[temperature]] in the upper layers of the troposphere where the influence from the surface and the [[boundary layer]] is negligible. Although the framework is formulated with the gradients of free tropospheric temperature, this phenomena occurs as a result of gradients and fluctuations in [[buoyancy]]. Density or buoyancy fluctuations in a stably stratified fluid lead to the formation of [[Gravity wave|gravity waves]].<ref name=":2" /> In the tropics, where Coriolis force is negligibly small, these [[Gravity wave|gravity waves]] prove to be very effective at smoothing out buoyancy gradients, in a process called gravity-wave adjustment or buoyant equalization.<ref>{{Cite journal |last=Bretherton |first=Christopher S. |last2=Smolarkiewicz |first2=Piotr K. |date=1989-03-15 |title=Gravity Waves, Compensating Subsidence and Detrainment around Cumulus Clouds |url=https://journals.ametsoc.org/view/journals/atsc/46/6/1520-0469_1989_046_0740_gwcsad_2_0_co_2.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=46 |issue=6 |pages=740–759 |doi=10.1175/1520-0469(1989)046<0740:GWCSAD>2.0.CO;2 |issn=0022-4928}}</ref> This effectively redistributes temperature between regions of precipitating convection and clear-sky region. Due to the speed with which the gravity-wave adjustment occurs, the WTG not only considers negligible horizontal buoyancy gradients but also negligibly small temporal gradients.<ref name=":4">{{Cite journal |last=Adames |first=Ángel F. |date=2022-08-01 |title=The Basic Equations under Weak Temperature Gradient Balance: Formulation, Scaling, and Types of Convectively Coupled Motions |url=https://journals.ametsoc.org/view/journals/atsc/79/8/JAS-D-21-0215.1.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=79 |issue=8 |pages=2087–2108 |doi=10.1175/JAS-D-21-0215.1 |issn=0022-4928}}</ref>
As, buoyancy is closely related to temperature (more specifically the [[virtual temperature]] and the virtual potential temperature) the framework is usually named Weak Temperature Gradient approximation.<ref name=":3" />
=== Equation
This framework can be approximated using scale analysis on the governing equations. Starting from the hydrostatic balance
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