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{{Short description|Meteorological concept}}
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>▼
{{Orphan|date=January 2025}}
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, and 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>▼
▲In [[atmospheric science]], the
▲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
Through a multitude of theoretical studies, modelling and observations, the WTG has been often applied to study [[Synoptic scale meteorology|synoptic]]- and [[Mesoscale meteorology|mesoscale]] phenomena in the tropics.▼
▲
== Physical Explanation ==▼
Free tropospheric temperature refers to the temperature found in the higher part of the troposphere where the influence from [[boundary layer]] effects is negligible. Although the framework is based on its gradients, this occurs as a result of gradients and fluctuations in [[buoyancy]]. Any stably stratified fluid which undergoes density or buoyancy fluctuations will lead to the formation of gravity waves.<ref name=":2" /> In the tropics, where Coriolis force is negligibly small, these 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 convective, precipitating regions and dryer regions. 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>▼
Buoyancy is closely related to temperature, more specifically virtual temperature and virtual potential temperature, leading to the name Weak Temperature Gradient.<ref name=":3" />▼
▲Free tropospheric temperature refers to the [[temperature
▲
=== Equation Derivation ===▼
This framework can be approximated using scale analysis on the governing equations. Starting from the hydrostatic balance
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</math>
* p: pressure
scale analysis suggests that the difference in pressure at two equal height <math>h</math> is ▼
* <math>\rho</math>: density
* g: gravitational acceleration
* z: height above surface
▲[[Scale analysis (mathematics)|scale analysis]] suggests that the difference (<math>\delta</math>) in pressure at two equal
<math>
\delta p \sim g h \delta \rho
</math><ref name=":2" />
These pressure differences can also be analyzed using the [[Navier–Stokes equations|Navier-Stokes]] momentum equation in the tropics with the [[Coriolis parameter]] <math>f \sim 0</math>
<math>
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</math>
* <math>\boldsymbol{u}</math> is the horizontal velocity component
Scale analysis now suggests that ▼
<math>
\frac{\delta \rho}{\rho}\sim \frac{\delta p}{p}\sim \frac{\delta \theta}{\theta}\sim \mathcal{F}_r
</math><ref name=":3" />
where <math>\mathcal{F}_r=\frac{U^2}{g h}</math> is the [[Froude number]], defined as the ratio of vertical inertial force to the gravitational force; <math>U</math> is a horizontal velocity scale. Whereas the same approach for extra-tropical regions would yield
<math>
\frac{\delta \rho}{\rho}\sim \frac{\delta\theta}{\theta}\sim \frac{\mathcal{F}_r}{R_o}
</math><ref name=":3" />
where <math>R_o=\frac{U}{f L}</math> is the [[Rossby number]] with L a characteristic horizontal length scale. This shows that for small Rossby
The pressure gradients mentioned above can be understood to be smoothed out by pressure gradient forces which in the tropics, unlike the mid-latitudes, are not balanced by Coriolis force and thus efficiently remove horizontal gradients.
== Applications ==
<math>
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</math>
* <math>\eta_d</math>: dry static energy
where Q is the diabatic heating from surface fluxes and latent heat effects, and <math>\omega</math> is the pressure velocity. This suggests that variations in a diabatic atmosphere allow for a formulation of equations for which temperature variations must follow a balance between vertical motions and diabatic heating.<ref name=":2" /><ref name=":4" />▼
* <math>\omega</math>: vertical pressure velocity
* Q: diabatic heating
▲where
There are two way to interpret this conclusion. The first, classical interpretation is that the large scale circulation creates conditions for atmospheric convection to occur. The alternate interpretation is that the surface fluxes and latent heat effects are the processes which control the large scale circulation. In this case, a heat source would cause a temperature anomaly which, in the WTG, would get smoothed out by gravity waves. Due to energetic constraints, this would lead to a large-scale vertical motion to cool the column.<ref name=":2" />▼
▲There are two
=== Models ===
The weak temperature gradient approximation is often
Many studies implemented the WTG constraint in radiative-convective equilibrium (RCE) models, by fixing the mean virtual temperature profile.<ref name=":0" /> Often this creates opposing results with either dry, non-precipitating
Bulk, single column models, can also be developed with the WTG. Although these models usually treat temperature prognostically while constraining the large-scale vertical motion, using the WTG approximation, large scale vertical motion becomes a diagnostic variable, dependent on static
Using the WTG framework, many different processes have been studied and better understood. These include, both synoptic processes such as the Walker Cell<ref>{{Cite journal |last1=Bretherton |first1=Christopher S. |last2=Sobel |first2=Adam H. |date=2002-10-15 |title=A Simple Model of a Convectively Coupled Walker Circulation Using the Weak Temperature Gradient Approximation |url=https://journals.ametsoc.org/view/journals/clim/15/20/1520-0442_2002_015_2907_asmoac_2.0.co_2.xml |journal=Journal of Climate |language=EN |volume=15 |issue=20 |pages=2907–2920 |doi=10.1175/1520-0442(2002)015<2907:ASMOAC>2.0.CO;2 |issn=0894-8755|url-access=subscription }}</ref> and the Madden Julian Oscillation<ref>{{Cite journal |last=Chikira |first=Minoru |date=2014-02-01 |title=Eastward-Propagating Intraseasonal Oscillation Represented by Chikira–Sugiyama Cumulus Parameterization. Part II: Understanding Moisture Variation under Weak Temperature Gradient Balance |url=https://journals.ametsoc.org/view/journals/atsc/71/2/jas-d-13-038.1.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=71 |issue=2 |pages=615–639 |doi=10.1175/JAS-D-13-038.1 |issn=0022-4928}}</ref> and also mesoscale processes such as, the diurnal cycle of convection,<ref>{{Cite journal |last1=Ruppert |first1=James H. |last2=Hohenegger |first2=Cathy |date=2018-06-15 |title=Diurnal Circulation Adjustment and Organized Deep Convection |url=https://journals.ametsoc.org/view/journals/clim/31/12/jcli-d-17-0693.1.xml |journal=Journal of Climate |language=EN |volume=31 |issue=12 |pages=4899–4916 |doi=10.1175/JCLI-D-17-0693.1 |issn=0894-8755|hdl=21.11116/0000-0000-0677-4 |hdl-access=free }}</ref> convective self-aggregation<ref>{{Cite journal |last1=Sessions |first1=Sharon L. |last2=Sugaya |first2=Satomi |last3=Raymond |first3=David J. |last4=Sobel |first4=Adam H. |date=2010-06-27 |title=Multiple equilibria in a cloud-resolving model using the weak temperature gradient approximation |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2009JD013376 |journal=Journal of Geophysical Research: Atmospheres |language=en |volume=115 |issue=D12 |doi=10.1029/2009JD013376 |issn=0148-0227}}</ref> and tropical cyclone formation.<ref>{{Cite journal |last1=Raymond |first1=David J. |last2=Sessions |first2=Sharon L. |date=March 2007 |title=Evolution of convection during tropical cyclogenesis |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2006GL028607 |journal=Geophysical Research Letters |language=en |volume=34 |issue=6 |doi=10.1029/2006GL028607 |issn=0094-8276|url-access=subscription }}</ref>
▲Many studies implemented the WTG constraint in radiative-convective equilibrium (RCE) models, by fixing the mean virtual temperature profile.<ref name=":0" /> Often this creates opposing results with either dry, non-precipitating results or heavily-precipitating states, depending on the stability of the constrained temperature profile.<ref>{{Cite journal |last=Wong |first=N. Z. |last2=Kuang |first2=Z. |date=2023-12-28 |title=The Effect of Different Implementations of the Weak Temperature Gradient Approximation in Cloud Resolving Models |url=https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2023GL104350 |journal=Geophysical Research Letters |language=en |volume=50 |issue=24 |doi=10.1029/2023GL104350 |issn=0094-8276}}</ref>
== References ==
▲Bulk, single column models, can also be developed with the WTG. Although these models usually treat temperature prognostically while constraining the large-scale vertical motion, using the WTG approximation, large scale vertical motion becomes a diagnostic variable, dependent on static energy and humidity. This solves the limitation of such models of understanding the distribution of precipitation as a prescribed vertical motion constrains precipitation. <ref name=":2" /><ref name=":1" />
{{Reflist}}
[[Category:Atmospheric temperature]]
[[Category:Tropical meteorology]]
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