Weak temperature gradient approximation: Difference between revisions

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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 |last1=Raymond |first1=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|url-access=subscription }}</ref><ref name=":1">{{Cite journal |last1=Sobel |first1=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>

InThe [[atmosphericassumption science]],of thehorizontal '''Weakhomogeneity Temperatureof Gradienttemperature approximation'''follows (WTG)from isobservations aof theoreticalfree frameworktropospheric usedtemperature toin the tropical regions as well as early work simplifyon the simplified equations governing tropical atmospheric dynamics and circulation. TheIt WTGis approximationunderstood assumesto thatoccur freeas [[Troposphere|tropospheric]]a temperatureresult inof the weak [[tropicsCoriolis force]] hasin negligiblethe horizontal (and temporal) gradients compared to its vertical gradienttropics.<ref name=":02">{{Cite journalbook |lasturl=Raymond |first=David Jhttps://www.cambridge. |last2=Zeng |first2=Xiping |date=2005org/core/books/clouds-04and-01climate/7B47159F7B050B71625111E40795D182 |title=ModellingClouds tropicaland atmosphericClimate: convectionClimate inScience's theGreatest contextChallenge of|date=2020 the|publisher=Cambridge weakUniversity temperaturePress gradient|isbn=978-1-107-06107-1 approximation|editor-last=Siebesma |urleditor-first=http://doi.wiley.com/10.1256/qj.03A.97 Pier |journal___location=QuarterlyCambridge Journal of the Royal Meteorological Society|editor-last2=Bony |languageeditor-first2=enSandrine |volumeeditor-last3=131Jakob |issueeditor-first3=608Christian |pageseditor-last4=1301–1320Stevens |doieditor-first4=10.1256/qj.03.97Bjorn}}</ref><ref name=":13">{{Cite journal |last=SobelCharney |first=AdamJule H. |last2=Bretherton |first2=Christopher SG. |date=20001963-1211-1501 |title=ModelingA TropicalNote Precipitationon inLarge-Scale aMotions Singlein Columnthe |url=https://journals.ametsoc.org/view/journals/clim/13/24/1520-0442_2000_013_4378_mtpias_2.0.co_2.xmlTropics |journal=Journal of Climatethe Atmospheric Sciences |language=EN |volume=1320 |issue=246 |pages=4378–4392607–609 |doi=10.1175/1520-04420469(20001963)013020<43780607:MTPIASANOLSM>2.0.CO;2 |issn=08940022-87554928|doi-access=free }}</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>

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 Explanationexplanation ==

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 phenomenaphenomenon occurs as a result of gradients and fluctuations in [[buoyancy]]. Density or buoyancy fluctuations in a stably stratified fluid lead to the formation of [[Gravitygravity wave|gravity waves]]s.<ref name=":2" /> In the tropics, where Coriolis force is negligibly small, these [[Gravitygravity wave|gravity waves]]s prove to be very effective at smoothing out buoyancy gradients, in a process called gravity-wave adjustment or buoyant equalization.<ref>{{Cite journal |lastlast1=Bretherton |firstfirst1=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|doi-access=free }}</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|url-access=subscription }}</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 Derivationderivation ===

This framework can be approximated using scale analysis on the governing equations. Starting from the hydrostatic balance

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=== Models ===

The weak temperature gradient approximation is often used in models with limited domains as a way to couple large-scale vertical motion and small scale diabatic heating. Generally, this has been done by neglecting horizontal free-tropospheric temperature variations (to first order), while explicitly retaining fluid dynamical aspects and diabatic processes.<ref>{{Cite journal |lastlast1=Sobel |firstfirst1=Adam H. |last2=Nilsson |first2=Johan |last3=Polvani |first3=Lorenzo M. |date=2001-12-01 |title=The Weak Temperature Gradient Approximation and Balanced Tropical Moisture Waves |url=https://journals.ametsoc.org/view/journals/atsc/58/23/1520-0469_2001_058_3650_twtgaa_2.0.co_2.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=58 |issue=23 |pages=3650–3665 |doi=10.1175/1520-0469(2001)058<3650:TWTGAA>2.0.CO;2 |issn=0022-4928|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 states or heavily-precipitating states, depending on the stability of the constrained temperature profile.<ref>{{Cite journal |lastlast1=Wong |firstfirst1=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|doi-access=free }}</ref> The WTG has also been used as a parametrization for large-scale motion in cloud-permitting models.<ref name=":4" />

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 stability and humidity. Due to the strong coupling between vertical motion and precipitation, the WTG approach allows the study of precipitation distribution, also in the bulk setup.<ref name=":2" /><ref name=":1" />

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 |lastlast1=Bretherton |firstfirst1=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 |lastlast1=Ruppert |firstfirst1=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 |lastlast1=Sessions |firstfirst1=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‐resolvingcloud-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 |lastlast1=Raymond |firstfirst1=David J. |last2=Sessions |first2=Sharon L. |date=March 2007-03 |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>

== References ==

{{Reflist}}

[[Category:Atmospheric temperature]]

[[Category:Tropical meteorology]]